Khan Academy · YouTube · 10:11
This 120-second clip introduces the graph of by first building a unit-circle interpretation of tangent. On the left, the instructor draws axes, a unit circle, and a terminal ray for an angle , labels the intersection point as , and writes the identity . He then explains that this same ratio is the slope of the ray, , giving a geometric way to understand tangent values. The right half of the screen already contains a large radian-labeled coordinate grid reserved for the eventual graph, but no tangent curve is plotted in this segment. The clip ends as the instructor cleans up the diagram and begins a table for different angles. This 120-second whiteboard segment introduces graphing by first building a small table of values from the unit circle. The presenter interprets tangent as the slope of the terminal ray, computes , explains using equal coordinates and the identity , and records . The right-hand coordinate grid remains blank during this excerpt. This 120-second whiteboard segment derives from unit-circle coordinates, fills a small table of tangent values, plots the first three points of , and then explains why the graph is not a line by showing that grows without bound as , culminating in a dashed vertical asymptote at . This 120-second whiteboard excerpt teaches the graph of by linking the unit-circle slope of a ray to the shape of the tangent curve. The instructor first completes the central branch between the vertical asymptotes at and , explaining that the slope tends to + near from the left and to -∞ near -π/2 from the right. He then continues past , showing that immediately after crossing the asymptote the ray’s slope is hugely negative, so the graph jumps to a new branch below the axis and rises again. Near the end, he introduces the example angle and begins rewriting it as , but the clip stops before the example is fully worked out. This 119-second whiteboard segment explains by treating tangent values as slopes of terminal rays on the unit circle. The speaker derives from a second-quadrant 45-45-90 triangle, then moves to and the next positive unit slope, plots corresponding points on the θ-y graph, and uses the unbounded behavior near to draw vertical asymptotes and state that the tangent graph repeats every π radians. This video segment introduces the graph of the tangent function, . It visually presents the characteristic repeating branches separated by vertical asymptotes. The narrator explains that the function is periodic, meaning the graph pattern continues indefinitely in both directions along the horizontal axis.
Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.
Generated from the video's visuals and explanation; not verbatim speech.
The clip opens with the function name written at the upper left while a large empty coordinate grid waits on the right. The immediate question is how to understand the tangent graph before actually plotting it.
To answer that, the instructor builds a small unit-circle diagram on the left: teal axes and a circle centered at the origin. This auxiliary picture is not the final graph; it is a geometric device for reading off tangent values from angles.
Next, a yellow ray is drawn from the origin into the first quadrant, and the angle between the positive -axis and the ray is labeled . This places the angle in standard position so its terminal side can meet the circle at a definite point.
The intersection point is then labeled . The key detail is the coordinate assignment: the horizontal coordinate is cosine and the vertical coordinate is sine. The speaker even audibly corrects an initial slip, reinforcing that the point is not .
With the point identified, the lesson writes the quotient identity . In this unit-circle setting, that means tangent is the ratio of the point’s -coordinate to its -coordinate.
The instructor then adds the slope interpretation . Because the ray starts at the origin and passes through , the rise-over-run of that ray is exactly the same ratio as . This is the central conceptual bridge of the clip: tangent can be seen as a slope, not merely as an algebraic fraction.
Finally, the extra annotations are cleared away and a new vertical line is drawn at the far left as the start of a table. The spoken transition indicates that the next stage will list several angles and compute the corresponding tangent values, but the table itself is not filled in before the clip ends.
The clip opens on a three-part whiteboard layout: a handwritten title , a two-column table headed and , a unit circle in the middle, and a blank Cartesian grid on the right. The purpose is to prepare for graphing the tangent function by first collecting a few reliable values.
The presenter starts with the easiest input, . On the unit circle the corresponding ray lies along the positive -axis, so there is no vertical change as the horizontal coordinate changes. Using the slope interpretation of tangent, this gives , and the pair is written into the table.
Next, the speaker explains the selection strategy: choose angles whose tangent values are easy to recognize, because those sample points will help reveal the overall shape of the graph of . This connects the geometric unit-circle reasoning to the numerical table that will later be transferred to the coordinate grid.
The next example is , described aloud as a 45-degree angle. A ray is drawn into the first quadrant, and a small right triangle is sketched to emphasize that the horizontal and vertical changes are equal. Since the slope is rise over run, equal changes imply slope , so .
The presenter reinforces the same conclusion algebraically by invoking . At , the sine and cosine values are the same, so their quotient is . The table is then updated with the row .
Finally, the speaker introduces and draws the corresponding ray below the positive -axis in the fourth quadrant. The written table entry records . The visual setup suggests a negative slope because the ray moves downward while progressing to the right, although the spoken explanation is cut off before it is fully completed.
The clip opens on a unit circle with the angle marked and a table for already listing and . The speaker identifies the relevant point on the circle: its cosine, the -coordinate, is , while its sine, the -coordinate, is . This establishes the coordinate inputs needed for the tangent calculation.
Using the definition , the video substitutes those two coordinates and obtains . The reasoning is that the magnitudes are equal but the signs are opposite, so the quotient is . By the end of this step, the missing table entry for is filled with .
Before moving to the graph, the presenter erases auxiliary construction marks from the unit circle so the same diagram can be reused. This cleanup does not change any values; it only prepares the board for the next stage, where numerical tangent values will be transferred onto a coordinate grid.
The right-hand grid is introduced as the graph of , with the horizontal axis labeled and the vertical axis labeled . The speaker then plots the first three known points directly from the table: , , and . These points anchor the beginning of the tangent curve.
Immediately after plotting them, the video warns against concluding that the graph is a straight line. The three points may look collinear at this coarse stage, but the speaker shifts attention to what happens as the angle increases toward . On the unit circle, the terminal ray becomes more and more vertical, so its slope becomes larger and larger positive.
At itself, the slope is undefined, but the limiting behavior as is that approaches . To represent this on the graph, a dashed vertical line is drawn at , identifying a vertical asymptote of the tangent function.
The screen is organized as a teaching board for : on the left are a handwritten title, a small table, and a cyan unit circle with yellow rays; on the right is a coordinate plane with horizontal axis θ and vertical axis y. This layout prepares a translation from geometric angle information to a function graph.
The instructor explains the central branch by tracking the slope of the ray on the unit circle. As the angle approaches from below, the ray becomes steeper and steeper in the positive direction, so the tangent value grows without bound; visually, the red curve rises toward the dashed line at .
He then asks what happens as the angle approaches -π/2. In the same slope picture, the ray becomes steeply negative, so the tangent value tends to -∞. On the graph, the left end of the branch drops toward the dashed line at .
At the boundary angle, the speaker states that the function is “not quite defined” and identifies a vertical asymptote. This turns the verbal limiting behavior into a clear graphical feature: the principal branch of tangent is trapped between two vertical asymptotes and passes through the origin.
Having finished the section over this part of the θ-axis, the instructor says the graph can keep going. The key move is to consider an angle just after crossing rather than stopping at the asymptote.
For an angle slightly larger than , the corresponding ray lies in the second quadrant. Its slope is therefore negative and very large in magnitude. This explains why the graph does not continue upward through the asymptote; instead it jumps to a new branch far below the x-axis just to the right of .
As θ increases further in that new interval, the slope becomes less and less negative, so the fresh branch rises. The visual logic is that each new period of tangent repeats the same pattern: unbounded growth near one asymptote, discontinuous reset, then renewed increase.
To anchor the next interval with a concrete example, the instructor marks an angle in the second quadrant and names it . He begins justifying the choice by rewriting it as , indicating that this angle lies a quarter-turn past the asymptote.
The clip ends while this example is still being set up. The mathematical point established so far is not a final numerical value for , but the method: use the unit-circle slope just beyond to explain why the tangent graph restarts on a lower branch and then increases.
The clip opens with the instructor locating on the unit circle by rewriting as and adding another . The visible whiteboard links three representations at once: a table of and , a unit circle with yellow terminal rays, and a gridded graph of .
He then explains why this angle matters geometrically. The terminal ray forms another -- triangle, so the horizontal and vertical distances have equal magnitude. Because the ray lies in the second quadrant, the x-distance is negative and the y-distance is positive. Under the clip’s slope interpretation of tangent, that sign pattern gives , matching the earlier slope seen at .
Next the angle is increased to . At that position the terminal ray is horizontal, so the slope returns to , giving . Increasing by another reaches the next special angle, visually consistent with , where the slope becomes positive again. These values are translated into plotted points on the right-hand graph.
The discussion then turns to what happens as the angle approaches . The speaker emphasizes that the slope becomes more and more positive and approaches positive infinity. In his words, a small movement in the x-direction corresponds to a very large movement upward in the y-direction, which is why the curve steepens dramatically near that boundary.
Using those behaviors, he sketches the tangent branch on the graph. The curve rises through the plotted points and shoots upward beside the vertical line at . He then states that the same pattern repeats every radians, so the graph continues in identical branches across neighboring intervals of length .
Finally, he draws dashed vertical asymptotes at the odd multiples of and uses them to separate the repeated branches. The closing visual message is that the graph of tangent is built from unit-circle slope values, has vertical asymptotes where the function becomes unbounded, and repeats periodically with period .
The video displays the equation alongside a unit circle diagram and a table of specific values. On the right, a coordinate plane shows the graph of the tangent function. The graph consists of multiple curved branches that pass through the origin and extend upwards and downwards towards infinity.
The narrator points out that the function is periodic. This means the pattern of the branches repeats at regular intervals. The graph shows these branches continuing infinitely to the left and right, separated by vertical dashed lines which represent asymptotes where the function is undefined.
The clip introduces the problem of becoming familiar with the graph of . Instead of plotting immediately, it first explains a geometric interpretation using the unit circle. The large right-hand grid is prepared for later graphing, but no curve is drawn in this excerpt.
A small coordinate system and circle are drawn on the left to represent the unit circle. This construction lets an angle be converted into a point in the plane, which then supports the definitions of sine, cosine, and tangent used in the rest of the clip.
The angle is shown with one side on the positive -axis and the other side as a ray from the origin. That ray intersects the unit circle, linking angle measure to a specific coordinate point.
The intersection of the terminal ray with the unit circle is labeled . The clip explicitly corrects the common swap error: the -coordinate is cosine and the -coordinate is sine.
The instructor states and writes . Within the unit-circle picture, this is the ratio of the point’s vertical coordinate to its horizontal coordinate. The clip does not explicitly mention the usual restriction .
Because the ray begins at the origin and passes through , the same ratio can be read as slope: change in over change in . This gives a direct geometric way to understand the magnitude and sign of tangent values as the angle changes.
After establishing the geometric meaning of tangent, the instructor cleans up the diagram and begins a table at the far left. The intended next step is to compare several angles with their tangent values, but the table entries are not shown within this 120-second clip.
This segment defines the working interpretation used throughout: for an angle , is read as the slope of the terminal ray drawn from the origin of the unit circle. A horizontal ray gives slope 0, an upward 45-degree ray gives slope 1, and a downward 45-degree ray gives slope -1.
When , the ray lies exactly along the positive -axis. As changes, does not change at all, so the slope is 0. The video records this as the first table entry.
Before drawing the full graph, the presenter chooses a few angles with especially simple tangent values. These sample points are entered into a table of versus and are intended to help infer the shape of the graph on the blank coordinate grid.
At , the unit-circle point has equal - and -coordinates. Moving the same amount horizontally and vertically means rise equals run, so the slope is 1. The presenter also notes that this corresponds to a 45-degree angle.
The video states the identity and applies it at . Because sine and cosine are equal there, their quotient is 1, matching the geometric slope argument.
The presenter then considers and draws the ray into the fourth quadrant. The table explicitly records the value , consistent with a downward-sloping ray of the same steepness as the case. The spoken justification begins but the clip ends before it is fully finished.
For the angle shown on the unit circle, the cosine is the -coordinate and the sine is the -coordinate . The video uses these exact coordinates as the starting data for computing tangent.
The tangent of an angle is computed as the ratio of sine to cosine. In this clip, that definition is applied to the unit-circle coordinates to turn geometric information into a function value.
Substituting the coordinates for gives a quotient of . This value is then written into the table as the entry corresponding to .
The graph is set up with horizontal axis and vertical axis . The first plotted points come directly from the table: , , and .
Even though the initial plotted points may suggest linear behavior, the video explicitly says the graph is not a line. The next step examines limiting behavior rather than connecting the points as if they determined a line.
As the angle approaches from below, the terminal ray on the unit circle becomes nearly vertical, so its slope becomes arbitrarily large positive. This is the intuitive basis for unbounded growth of tangent near .
Because becomes undefined at and tends to infinity on approach, the graph is marked with a dashed vertical asymptote at .
Between and , the graph is drawn as one increasing branch through the origin. The instructor obtains this shape from the unit-circle slope picture: near the slope becomes unbounded positive, and near -π/2 it becomes unbounded negative.
The clip explicitly says the function is “not quite defined” at the boundary and marks those locations with vertical dashed lines. These asymptotes separate one branch of the tangent graph from the next.
The lesson repeatedly interprets tan θ as the slope of the ray making angle θ with the positive x-axis. Steeper positive rays give larger positive tangent values; steep negative rays give large negative tangent values.
Immediately after crossing , the ray enters the second quadrant, where its slope is negative and very large in magnitude. Therefore the tangent graph does not continue upward through the asymptote; it restarts on a new branch below the x-axis and then rises as θ increases.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
At the upper left, the handwritten title/function label is .
The speaker says he wants to get familiarity with the graph of tangent of theta.
The dependent variable is the value of the tangent function at angle ; this is the function whose graph is being introduced.
is an angle in radians on the coordinate grid; is real where is defined.
The speaker says, "So let's say that's a y-axis. This is my x-axis."
A teal vertical axis and horizontal axis are drawn on the left side as a unit-circle coordinate system.
-axis
Horizontal axis of the auxiliary unit-circle diagram; it also corresponds to the positive initial side of the angle .
Real line in the auxiliary diagram.
The speaker identifies the vertical line as the y-axis while drawing the unit circle setup.
A teal vertical axis is drawn on the left side.
-axis
Vertical axis of the auxiliary unit-circle diagram; it supplies the sine coordinate of the point on the circle.
Real line in the auxiliary diagram.
The speaker says, "if I have an angle, an angle theta, where one side is the positive x-axis".
A yellow ray from the origin forms an angle with the positive -axis, and the angle is marked .
An angle measured from the positive -axis to a ray intersecting the unit circle.
Angles in radians; in this clip the drawn example lies in the first quadrant.
The speaker corrects himself and says the coordinate is the cosine of theta.
The point on the circle is labeled .
The -coordinate of the intersection point of the terminal ray with the unit circle.
For the unit circle, ; here it is shown as the horizontal leg of the right triangle.
The speaker says the coordinate there is the sine of theta.
The point on the circle is labeled .
The -coordinate of the intersection point of the terminal ray with the unit circle.
For the unit circle, ; here it is shown as the vertical leg of the right triangle.
The speaker says, "we know that tangent of theta is the same thing as the sine of theta over the cosine of theta."
The handwritten equation appears below the title.
Tangent is defined as sine divided by cosine, equivalently the ratio of the point’s -coordinate to its -coordinate on the unit circle.
Valid when ; the clip does not explicitly state the excluded angles.
The speaker says this is going to be your change in y.
The handwritten slope expression includes in the numerator.
Change in the vertical coordinate from the origin to the point on the unit circle; in this setup it equals .
Shown for the drawn first-quadrant ray; generally a vertical displacement.
The speaker says this is over change in x.
The handwritten slope expression includes in the denominator.
Change in the horizontal coordinate from the origin to the point on the unit circle; in this setup it equals .
Shown for the drawn first-quadrant ray; generally a horizontal displacement.
The intersection point is labeled .
The speaker states that where the ray intersects the unit circle, the coordinates are cosine theta comma sine theta.
Coordinates of the point where the terminal side of angle meets the unit circle.
Points on the unit circle .
Top-left handwritten title reads .
The speaker refers to the shape of the graph of y is equal to tangent of theta.
The plotted function whose value at angle θ is the tangent of θ.
θ is an angle in radians; y is the corresponding tangent value.
The table header uses θ.
The speaker repeatedly says “if theta is...” while choosing example angles.
Input angle used to evaluate the tangent function and to draw a ray on the unit circle.
Angles shown in this clip include 0, , and -π/4 radians.
The speaker says, "So what I hope to do in this video is get some familiarity with the graph of tangent of theta."
The title is written at the upper left.
This clip introduces the task of understanding the tangent graph by connecting it to the unit circle rather than by plotting points immediately. The visible target function is , and the large coordinate grid on the right is prepared for later graphing.
The clip is introductory.
No explicit domain restriction is stated in this segment.
The speaker says he will set up a little unit circle so we can visualize what the tangent of various thetas are.
Teal axes and a circle centered at the origin are drawn on the left side.
The instructor draws an auxiliary coordinate system and a circle centered at the origin to represent the unit circle. This construction is used as a geometric reference for interpreting through coordinates and slope.
The circle is presented as a unit circle.
The drawing is an auxiliary visualization beside the main graph grid.
The speaker says, "if I have an angle, an angle theta, where one side is the positive x-axis".
A yellow ray is drawn from the origin into the first quadrant, forming an angle with the positive -axis.
The angle is represented in standard position: one side lies along the positive -axis and the other side is a ray from the origin. The clip uses this ray to connect an angle to a point on the unit circle.
One side of the angle is the positive -axis.
The other side is a ray from the origin.
The speaker says the coordinates where the ray intersects the unit circle are cosine theta comma sine theta, after correcting an initial slip.
The point is labeled .
When the terminal side of intersects the unit circle, the intersection point’s horizontal coordinate is and its vertical coordinate is . This is the bridge from angle measure to coordinate geometry used later for tangent.
The point lies on the unit circle.
The angle is measured from the positive -axis to the terminal ray.
The speaker says tangent of theta is the same thing as the sine of theta over the cosine of theta.
The handwritten equation is shown.
The clip states the quotient identity for tangent. In the unit-circle picture, this means tangent is the ratio of the point’s -coordinate to its -coordinate.
The identity is used for the angle shown on the unit circle.
The clip does not explicitly mention the restriction .
The speaker says taking the y-coordinate over the x-coordinate is essentially the slope of this line, change in y over change in x.
The handwritten slope expression is added beneath the tangent identity.
Because the ray starts at the origin and passes through , the ratio is also the rise-over-run slope of that ray. This gives a geometric way to think about the size and sign of .
The line/ray passes through the origin.
The referenced point is the unit-circle intersection .
The clip does not explicitly discuss the undefined case when the run is zero.
The speaker says, "So now let's make a table," and begins discussing various thetas and what tangent of theta would be.
The unit-circle annotations are cleaned up, and a new vertical yellow line is drawn at the far left as the start of a table.
The table headings and entries are not yet visible by the end of the clip.
After establishing that tangent can be read from the unit circle as a coordinate ratio or slope, the instructor clears space and begins a table. The intended next step is to list several angles and determine the corresponding values.
This is only the beginning of the table-making step within the clip.
No completed table entries appear before 120 seconds.
The speaker explains tangent values by asking what the slope of the ray is.
Each chosen angle is represented by a ray from the origin of the unit circle.
In this clip, the presenter treats as the slope of the ray making angle with the positive x-axis on the unit circle. The reasoning is visual: compare how much the ray rises vertically for a given horizontal run.
θ is measured from the positive x-axis.
The ray is drawn on the unit circle diagram.
A cyan unit circle with x- and y-axes is shown, and yellow rays are drawn from the origin for selected angles.
The speaker points to the ray while discussing its slope.
The method shown is to pick convenient angles, draw the corresponding terminal ray on the unit circle, and read off the tangent value from the ray’s slope or from the equality of coordinate changes.
A unit circle is available.
Selected angles should have easily recognized coordinates or slopes.
A two-column table labeled θ and tan θ is built on the left.
The speaker says these values will help us think about the shape of the graph of .
Before sketching the full graph, the video records sample input-output pairs in a table. This turns the geometric unit-circle information into numerical points that can later be transferred to the coordinate grid.
Useful when the goal is to infer the overall shape of a function graph from selected values.
The speaker says if theta is zero radians, the slope is zero because as x changes, y doesn't change at all.
The first table row records and .
At , the terminal ray lies along the positive x-axis, so there is no vertical change as the horizontal coordinate changes. Therefore the slope, and hence the tangent value, is 0.
radians.
The speaker says for pi over four radians, your x-coordinate and your y-coordinate is the same, so the slope of this ray is going to be equal to 1.
The table records and .
For , the unit-circle point has equal x- and y-coordinates, so moving horizontally and vertically by the same amount gives slope 1. The presenter also states that and are the same, so their ratio is 1.
radians.
Equivalently, the sine and cosine values are equal and nonzero.
The speaker states that tangent of theta is the same thing as sine of theta over cosine of theta.
The equation is written on screen.
The video does not explicitly state the condition in this clip.
For the angle shown on the unit circle, .
is an angle in standard position on the unit circle.
The point on the unit circle determined by has coordinates .
The clip leaves unstated the usual requirement for the quotient to be defined.
Stated for the angle under discussion; no explicit universal quantifier is spoken in this clip.
The speaker says the y-coordinate over the x-coordinate is essentially the slope of this line, change in y over change in x.
The slope expression is written alongside the tangent identity.
The spoken wording uses "this line" while the drawn object is a ray from the origin; the mathematical meaning is the slope of the terminal side.
In the unit-circle setup, is the slope of the ray from the origin through , namely .
The ray begins at the origin.
It intersects the unit circle at .
Slope is computed as change in divided by change in .
Presented for the example angle and the associated ray.
The speaker says if theta is zero radians, the slope is zero because as x changes, y doesn't change at all.
The ray for lies along the positive x-axis.
If the terminal ray for θ lies along the positive x-axis, then its slope is 0, so .
radians.
The ray is drawn from the origin along the positive x-direction.
For the specific angle shown in the clip.
The speaker says your x-coordinate and your y-coordinate is the same ... so the slope of this ray ... is going to be equal to 1.
He adds that sine of theta over cosine of theta are the same thing, so you get 1.
When the x- and y-coordinates on the unit circle are equal and nonzero, the ray’s slope is 1, equivalently , so .
The point lies on the unit circle.
The x-coordinate equals the y-coordinate.
The common cosine value is nonzero.
Applied in the clip to .
The left table visibly contains the row , .
A fourth-quadrant ray is drawn for the negative angle.
The spoken justification is cut off before completion, but the written result is clear.
The clip records .
radians.
For the specific angle shown in the table.
At 01:19.000 the speaker asks what happens as the angle gets closer and closer to .
At 01:35.000 the speaker says the ray is getting closer and closer to approaching the vertical.
At 01:40.000 the speaker says its slope is getting more and more and more positive.
At 01:44.000 the speaker says at the slope is undefined but is approaching infinity.
As approaches from below, the tangent value grows without bound, so the graph has a vertical asymptote at .
is measured in radians.
The approach is from values less than .
For tending to from the left.
At 01:53.000 the speaker says, "So as you get closer and closer to pi over two ... I'm going to draw essentially a vertical asymptote right over here at pi over two".
A dashed vertical line is drawn at on the right-hand graph.
The graph of has a vertical asymptote at .
Graphing .
Angles in radians.
At the specific value shown in this clip.
At 00:00:38-00:00:42 the speaker says, “So once again, not quite defined right over there, we have a vertical asymptote.”
The clip states non-definition at the marked location but does not provide a formal algebraic proof in this excerpt.
At the boundary angle being discussed, the tangent graph is not defined and the curve has a vertical asymptote there.
The angle is the boundary location shown on the graph near -π/2 or .
The graph is being interpreted through the unit-circle slope picture used in the clip.
Local statement about the specific boundary angle under discussion in the clip.
At 00:00:13-00:00:18 the speaker says the slope gets closer and closer to and is getting closer and closer to infinity.
The central red branch rises sharply toward the dashed line at .
The clip expresses the idea verbally and visually rather than writing a limit notation.
As the angle approaches from below, the slope of the corresponding ray, and hence the tangent value, grows without bound positively.
θ is approaching from values less than .
The tangent value is identified with the slope of the ray in the unit-circle picture.
One-sided limiting behavior near .
At 00:00:18-00:00:36 the speaker asks what happens as the angle gets closer and closer to negative and answers that the slope is approaching negative infinity.
The central red branch descends sharply toward the dashed line at .
The clip expresses the idea verbally and visually rather than writing a limit notation.
As the angle approaches -π/2 from above, the slope of the corresponding ray becomes more and more negative without bound.
θ is approaching -π/2 from values greater than -π/2.
The tangent value is identified with the slope of the ray in the unit-circle picture.
One-sided limiting behavior near -π/2.
At 00:01:02-00:01:15 the speaker says that right after crossing the slope is hugely negative.
At 00:01:19-00:01:24 the speaker says the graph jumps back down here and is hugely negative again.
A new branch is started below the x-axis just to the right of the asymptote at .
The clip does not finish deriving the full next branch before ending.
Just after the angle crosses , the corresponding slope is very large negative, so the graph of tangent drops to a new branch below the axis and then rises as θ increases.
θ is slightly greater than .
The same unit-circle slope interpretation is being continued from the previous branch.
Local statement about angles just to the right of .
The speaker says, “So at three pi over four, we have a slope of negative one.”
The table already lists -π/4 with , and the speaker compares to that slope.
.
θ is measured in radians.
The angle is on the unit circle.
Tangent is interpreted as the slope of the terminal ray.
For the specific angle .
The speaker moves from the point to and then to slope .
The screen shows , then , then .
The terminal ray of angle meets the unit circle at a point whose coordinates are cosine theta and sine theta.
Unit-circle definition of sine and cosine, stated by the speaker.
Tangent is rewritten as the ratio of the point’s -coordinate to its -coordinate.
Quotient identity for tangent, written and spoken in the clip.
Because the ray starts at the origin, that same coordinate ratio is the rise over run of the ray.
Definition of slope as change in divided by change in , explained verbally and written on screen.
The clip connects three equivalent viewpoints: the unit-circle coordinates, the quotient identity for tangent, and the slope of the terminal ray.
The speaker repeatedly computes tangent by asking what the slope of the ray is.
Rays and small right triangles are drawn on the unit circle for selected angles.
Choose an angle θ and draw its terminal ray from the origin of the unit circle.
This is the setup shown on the left diagram.
Read the ray’s vertical change relative to its horizontal change, i.e. its slope.
The speaker explicitly interprets tangent as the slope of the ray.
For , the ray is horizontal, so y does not change as x changes.
Stated directly in the audio and matched by the drawn ray.
For , the x- and y-coordinates are the same, so the rise equals the run and the slope is 1.
Explained verbally using equal coordinates and the identity .
For , the ray goes downward while moving right, giving a negative slope of magnitude 1.
The table entry is written explicitly; the verbal explanation begins but the clip ends before it is fully completed.
The sample values tabulated in the clip are , , and .
The speaker says tangent of theta is going to be equal to 1, or sine of theta over cosine of theta, they're the same thing, so you're going to get 1.
Rewrite tangent as the quotient of sine and cosine.
This identity is stated aloud by the presenter.
At , the sine and cosine values are the same.
The speaker says the x-coordinate and y-coordinate are the same and mentions √ for both.
Dividing equal nonzero quantities gives 1.
Direct arithmetic consequence of equality of numerator and denominator.
This confirms without relying only on the visual slope argument.
At 00:00.000 the speaker says sine and cosine of the angle are opposites.
At 00:04.000 the speaker gives cosine as and at 00:10.000 gives the -coordinate as .
At 00:18.000 the speaker computes tangent as sine over cosine and obtains .
From the unit-circle point for the angle, the -coordinate is positive and the -coordinate is negative .
Directly stated in the audio at 00:04.000-00:17.000.
The tangent is formed as the ratio of sine to cosine.
Stated in the audio at 00:18.000-00:23.000.
Substituting the two coordinates gives a quotient of .
Algebraic simplification of the displayed ratio; the final value is also written into the table by 00:45.000.
The tangent value for the angle is .
At 01:19.000 the speaker asks what happens as the angle gets closer to .
At 01:35.000 the speaker says the ray approaches vertical.
At 01:40.000 the speaker says the slope becomes more and more positive.
At 01:44.000 the speaker says the slope at is undefined but approaching infinity.
At 01:53.000-02:00.000 a dashed vertical asymptote is drawn at .
The angle is increased toward from smaller values.
Stated in the audio beginning at 01:19.000.
On the unit circle, the terminal ray approaches the vertical direction.
Stated in the audio at 01:35.000-01:40.000.
Because tangent is interpreted as the slope of the terminal ray, the slope grows without bound as the ray nears vertical.
Stated in the audio at 01:40.000-01:52.000.
The unbounded growth is represented on the graph by a dashed vertical line at .
Drawn and described in the audio at 01:53.000-02:00.000.
The tangent graph develops a vertical asymptote at because the corresponding ray becomes vertical and its slope tends to infinity.
At 00:00:02-00:00:18 the speaker explains approach to infinity near .
At 00:00:18-00:00:36 the speaker explains approach to negative infinity near -π/2.
The red curve is drawn between the two dashed asymptotes and through the origin.
Start from the unit-circle picture: a ray from the origin represents the angle θ, and the lesson treats tan θ as the slope of that ray.
Explicitly stated in the audio at 00:00:13-00:00:18 and supported by the left-hand diagram.
As θ moves closer to , the ray becomes steeper and steeper in the positive direction, so the slope tends to +.
Directly stated by the speaker at 00:00:13-00:00:18 and shown by the rising right end of the red branch.
As θ moves closer to -π/2, the ray becomes steeper and steeper in the negative direction, so the slope tends to -∞.
Directly stated by the speaker at 00:00:18-00:00:36 and shown by the falling left end of the red branch.
Therefore the graph on the interval between -π/2 and is a single increasing branch passing through the origin and bounded by vertical asymptotes at both ends.
This is the visual conclusion assembled from the preceding slope descriptions and the drawn curve.
The central branch of is obtained by translating the unbounded positive and negative slope behavior near ± into a graph with vertical asymptotes at those angles.
At 00:00:51-00:01:01 the speaker says the graph can keep going beyond the displayed section.
At 00:01:02-00:01:15 the speaker analyzes the angle just after crossing and calls the slope hugely negative.
At 00:01:19-00:01:24 the speaker says the graph jumps back down and is hugely negative again.
A new red branch begins below the axis to the right of the asymptote at .
The derivation is only partially completed before the clip ends.
Consider an angle just larger than on the unit circle.
Explicitly introduced by the speaker at 00:01:02-00:01:09.
The corresponding ray now points into the second quadrant, where its slope is negative and very large in magnitude.
Stated by the speaker at 00:01:10-00:01:18 as “hugely negative.”
Thus the tangent graph does not continue upward through the asymptote; instead it reappears on a new branch far below the x-axis.
Stated by the speaker at 00:01:19-00:01:24 and shown by the newly drawn lower branch.
As θ continues to increase past , the slope becomes less negative, so the new branch rises.
Stated by the speaker at 00:01:24-00:01:33.
Immediately after , the tangent graph jumps to a new branch starting from very negative values and then increases as the angle grows.
The speaker derives from plus and then explains the quadrant signs.
The unit circle shows a second-quadrant ray and the handwritten label .
The speaker rewrites as and adds another to identify the angle .
Arithmetic on radian measures.
He notes that this angle forms another 45-45-90 triangle, so the horizontal and vertical distances have the same magnitude.
Special right-triangle geometry on the unit circle.
Because the terminal ray lies in the second quadrant, the x-distance is negative and the y-distance is positive.
Quadrant sign convention on the coordinate plane.
Equal magnitudes with opposite signs give a ratio of -1, matching the slope previously seen at -π/4.
Tangent is treated as slope, or rise over run, on the terminal ray.
The video concludes that the tangent value at is -1.
The speaker moves from to π and then by another , stating the slope returns to 0 and then to positive 1.
Plotted points appear at , and near , .
The exact symbolic label is not separately written in the visible frames, though the spoken increment and plotted point support it.
Starting from the just-derived point, the graph has a negative unit slope value.
Previous derivation in the clip.
As the angle increases to π, the terminal ray lies horizontally and the slope returns to zero.
Geometric position of the terminal ray on the unit circle.
Increasing by another puts the ray in the third quadrant, where both coordinate changes have the same sign, restoring a positive unit slope.
Symmetry of the unit circle and the slope interpretation of tangent.
The plotted tangent values progress from -1 at to 0 at π and then to 1 at .
The speaker says the slope approaches positive infinity near , then says the pattern continues every pi radians and draws asymptotes.
A steep branch rises toward a dashed vertical line at , and similar branches are drawn in other π-length intervals.
Near , a small movement in x corresponds to a very large movement upward in y, so the tangent value grows without bound.
Spoken description of steepening slope and visible curve behavior.
The graph is separated at that angle by a dashed vertical line.
Visual construction in the clip.
The same branch shape is then reproduced in adjacent intervals of length π.
Explicit spoken periodicity claim plus repeated drawing.
The clip uses unbounded behavior near to justify drawing vertical asymptotes and repeating the tangent graph every π radians.
The left table accumulates rows (0,0), (,1), and (-π/4,-1).
The speaker says he is picking values that are very easy for us to think about and that will help form the shape of the graph.
The final verbal explanation for the -π/4 case is incomplete because the clip ends mid-sentence.
Select convenient angles, determine their tangent values from the unit circle, and record them in a table to prepare for graphing .
A unit circle diagram is available.
The target function is .
Chosen angles are 0, , and -π/4.
Compute and tabulate corresponding tangent values.
The ray at 0 radians is horizontal, so its slope is 0.
Stated directly in the audio and written in the table.
At the coordinates are equal, so rise equals run and the slope is 1.
Explained using both the unit-circle picture and .
At -π/4 the ray points into the fourth quadrant, giving a negative slope of magnitude 1.
The result is written in the table; the spoken justification begins but is cut off.
The table contains , , and .
Each entry matches the drawn ray direction on the unit circle and the slope-based interpretation used throughout the clip.
By 00:45.000 the table contains , , and .
At 00:46.000 the speaker says, "let's just start plotting a few of these points".
Between 00:58.000 and 01:11.000 the three corresponding points are plotted on the right-hand graph.
Use the known tangent values in the table to begin graphing .
Horizontal axis is and vertical axis is .
Place the first points of the tangent graph on the coordinate grid.
Plot the point corresponding to tangent of negative .
Given by the table and stated in the audio at 01:07.000.
Plot the origin point corresponding to tangent of zero.
Given by the table and stated in the audio at 00:58.000.
Plot the point corresponding to tangent of .
Given by the table and stated in the audio at 01:03.000.
The graph starts with the three points , , and .
These points match the completed table entries visible by 00:45.000 and the plotted marks visible by 01:11.000.
At 00:01:33-00:01:52 the speaker introduces an angle and says, “Let’s say that this angle right over here is .”
At 00:01:53-00:01:59 the speaker begins explaining why was chosen: “because that is plus .”
Handwritten appears below the table near the end of the clip.
The example is only begun; no final tangent value or completed graph point for is shown within this clip.
Choose a sample angle just beyond to continue explaining the next branch of the tangent graph.
The current discussion is on angles just after crossing .
The speaker selects the angle .
Use to illustrate how the tangent graph behaves in the next interval after the asymptote.
The speaker first notes that the graph can keep going beyond the central section.
Audio at 00:00:51-00:01:01.
He then considers an angle just after crossing and observes that its slope is hugely negative.
Audio at 00:01:02-00:01:18.
To make the next example concrete, he names the angle .
Audio at 00:01:47-00:01:52 and handwritten near the end.
He begins to justify the choice by rewriting as plus .
Audio at 00:01:53-00:01:59.
The clip identifies as the next example angle and starts explaining it as , but the worked consequence is not completed within the provided duration.
No completed numerical check or final plotted value for is visible in this excerpt.
The speaker names the angles , π, and the next increment and states their slopes.
Red points are added on the right graph at (,-1), (π,0), and approximately (,1).
The third plotted angle is visually consistent with , but the exact label is not separately written on screen.
Use the unit-circle slope idea to place several points of on the graph beyond .
The table already contains (-π/4,-1), (0,0), and (,1).
The speaker identifies by adding to .
Tangent is interpreted as slope of the terminal ray.
Determine and plot the tangent values at , π, and the next increment.
The second-quadrant 45-45-90 setup gives equal magnitudes with x negative and y positive, so the slope is -1.
Unit-circle geometry and quadrant signs.
This ordered pair is plotted on the θ-y graph.
Direct translation from angle and tangent value to graph coordinates.
At π the terminal ray is horizontal, so the slope returns to zero.
Geometric position on the unit circle.
The point at is plotted on the horizontal axis.
Graphing the computed tangent value.
After increasing by another , the slope is positive one again.
Spoken continuation of the unit-circle pattern and visible plotted point.
The example produces the plotted points (,-1), (π,0), and approximately (,1).
The plotted red points match the spoken slope values and align with the grid labels on the θ-axis and y-axis.
A large white coordinate grid occupies the right half of the screen, with x-axis labels from to in multiples of and y-axis labels from to .
The handwritten title appears at the upper left.
Large Cartesian grid on the right
Axis tick labels in radians
Handwritten function title
The title is written at the upper left.
The right-side grid remains empty of a plotted curve during this clip.
The right grid stays fixed as the eventual graphing area.
No tangent curve is drawn yet.
The visual layout separates the conceptual workspace on the left from the future graphing area on the right, signaling that the lesson will build understanding before plotting.
Teal axes are drawn on the left, followed by a circle centered at their intersection.
The speaker says he is setting up a little unit circle to visualize tangent values.
Teal vertical axis
Teal horizontal axis
Circle centered at the origin
Axes are added on the left side.
A circle is drawn around the origin.
The main right-hand grid is unchanged.
The circle is presented as a unit circle.
The new left-side diagram serves as a geometric model for converting an angle into coordinates and then into a tangent value.
A yellow ray is drawn from the origin into the first quadrant, and the angle between the positive -axis and the ray is marked .
The speaker describes an angle theta where one side is the positive x-axis.
Positive -axis
Yellow terminal ray
Angle mark
A ray is added from the origin.
The angle label appears between the axes and the ray.
The circle remains centered at the origin.
The initial side stays on the positive -axis.
The picture instantiates the abstract angle as a concrete ray, preparing to locate its intersection with the unit circle.
The intersection point is labeled .
Dashed guide lines form a right triangle from the origin to the point, and is written.
Point on the circle
Dashed horizontal and vertical guide segments
Right triangle with legs along the axes directions
Written ratios and
The intersection point receives coordinate labels.
Dashed segments create a right triangle.
Slope notation is added beneath the tangent identity.
The point remains on the unit circle.
The ray still starts at the origin.
The visual triangle makes the algebraic identities geometric: the vertical leg represents or , the horizontal leg represents or , and their ratio is the slope, hence .
The detailed annotations around the unit circle are erased, leaving a simpler circle-and-ray sketch.
A new vertical yellow line is drawn at the far left as the beginning of a table.
The speaker says, "So now let's make a table."
The table contents are not yet written by the end of the clip.
Simplified unit-circle sketch
New vertical yellow line at far left
Extra labels and dashed guides are removed.
A fresh column-like line is drawn to begin a table.
The unit circle remains as the conceptual reference.
The right-hand graph grid is still unused.
The instructor transitions from explaining why tangent has a geometric meaning to organizing specific angle-value pairs for graphing.
The screen is split into three regions: a left handwritten table, a middle unit circle, and a right blank coordinate grid.
The left table headers are θ and tan θ.
Left two-column table labeled θ and tan θ
Middle unit circle with x- and y-axes
Right Cartesian grid with x-axis marked from -2π to 2π and y-axis from -4 to 4
Title
The table is filled row by row as angles are discussed.
Yellow rays are added to the unit circle for each chosen angle.
The unit circle remains centered at the origin.
The right graph stays blank throughout this clip.
The visual organization separates numerical recording, geometric reasoning, and eventual graph plotting.
A yellow ray lies along the positive x-axis of the unit circle.
The first table row records 0 under θ and 0 under tan θ.
Positive x-axis ray
Table row (0,0)
The first input-output pair is entered into the table.
The ray coincides with the horizontal axis.
This visually demonstrates zero vertical change, hence slope 0 and .
A yellow ray is drawn into the first quadrant at 45 degrees, and a small right triangle is sketched near it.
The table later records under θ and 1 under tan θ.
The exact handwritten angle label on the circle is less clear than the spoken value and the table entry.
First-quadrant ray
Small right triangle indicating equal horizontal and vertical legs
Table row (,1)
A new ray is added above the x-axis.
The corresponding tangent value 1 is written in the table.
The ray starts at the origin and meets the unit circle in the first quadrant.
Equal leg lengths show slope 1, matching .
A yellow ray is drawn into the fourth quadrant below the positive x-axis.
The table records -π/4 under θ and -1 under tan θ.
The spoken explanation of why the value is negative is incomplete because the clip ends mid-sentence.
Fourth-quadrant ray
Table row (-π/4,-1)
A downward-sloping ray is added.
The tangent value -1 is entered in the table.
The ray still originates at the center of the unit circle.
The downward direction relative to rightward motion indicates a negative slope, consistent with .
At 00:31.000 the speaker says, "let me clean this up a little bit because I want to keep reusing my unit circle".
The extra construction marks around the unit circle are erased, leaving a cleaner circle and axes.
Unit circle
Terminal ray for
Coordinate axes
Temporary construction marks are removed.
The unit circle is preserved for reuse.
The underlying angle value and tangent result remain unchanged.
This visual cleanup separates the computation stage from the upcoming graphing stage.
Three points appear on the right-hand grid at , , and with respectively.
The speaker names each value as it is plotted between 00:58.000 and 01:11.000.
Right-hand coordinate grid
Point
Point
Point
Points are added one by one onto the graph.
The horizontal axis is identified as and the vertical axis as .
The scale markings in multiples of remain fixed.
The graph is initialized using exact values already computed from the unit circle.
A dashed vertical line is drawn at on the right-hand graph.
At 01:53.000 the speaker says he is drawing "essentially a vertical asymptote right over here at pi over two".
Dashed vertical line
Graph grid
Label on horizontal axis
A new vertical guide appears at .
The graphing discussion shifts from discrete points to limiting behavior.
Previously plotted points remain in place.
The dashed line visually encodes that becomes unbounded as approaches .
The speaker initially says the coordinates are the sine of theta, then immediately corrects himself: "Sorry, the x coordinate is the cosine of theta."
One may mistakenly think the -coordinate of the unit-circle point is .
The clip explicitly corrects this: the -coordinate is and the -coordinate is , so the point is .
The speaker emphasizes that taking y over x is essentially the slope of the line.
is written next to .
Students may treat purely as an algebraic formula and miss its geometric meaning.
In this unit-circle setup, the same ratio is the slope of the terminal ray, which helps visualize how tangent changes with angle. This interpretation is directly supported by the clip.
The presenter repeatedly uses the slope of the ray rather than only symbolic manipulation.
Each value is justified geometrically on the unit circle.
Students may treat as purely an algebraic ratio and miss its geometric meaning as a slope on the unit circle.
This clip emphasizes that tangent can be read directly from the steepness and direction of the terminal ray.
The ray for -π/4 is drawn below the x-axis.
The table entry is -1.
The video does not finish the verbal explanation before the clip ends.
Learners may think a negative angle merely flips the picture without affecting the sign of the slope.
Here the negative angle produces a downward-sloping ray, which is why the tangent value is negative.
At 01:12.000 the speaker says, "if you just saw that you might say oh maybe it's some type of a line but we'll see very clearly it's not a line".
After plotting only a few points, one may think the tangent graph is a line.
The video explicitly states that the graph is not a line, and then explains unbounded growth near .
At 00:00:38-00:00:42 the speaker says the function is “not quite defined right over there, we have a vertical asymptote.”
At 00:01:19-00:01:24 the speaker says the graph “jumps back down here” after crossing .
The graph is broken by vertical dashed lines and a new branch starts separately on the other side.
One might think the graph should continue smoothly upward past instead of breaking and restarting on another branch.
The clip explicitly marks the boundary as undefined and shows a vertical asymptote, then demonstrates that just after the graph jumps to a very negative branch rather than continuing through the asymptote.
At 00:01:10-00:01:18 the speaker emphasizes that just after crossing the slope is “hugely negative.”
The new branch is drawn below the x-axis to the right of the asymptote.
Because the previous branch was rising to +, one may expect nearby angles just beyond also to give large positive tangent values.
The unit-circle slope picture shows that immediately after the ray lies in the second quadrant, so its slope is large negative, and the graph restarts from below.
The speaker repeatedly derives new values from unit-circle geometry instead of reading only the table.
The unit circle is used to explain and later angles.
One might think tangent values can only be taken directly from the listed table rows.
The video shows that new values such as , , and the next positive-one value are obtained by extending the unit-circle reasoning, not merely by copying the table.
The speaker says the slope approaches positive infinity near .
The graph is broken by dashed vertical lines rather than passing through them.
A learner may imagine that the tangent graph attains a finite y-value exactly at the dashed vertical line.
In the clip, angles such as are shown as boundaries where the curve grows without bound and the graph is separated by dashed asymptotes, not as ordinary points on the curve.
The speaker says he will set up a unit circle in order to visualize tangent values for the graph of tangent.
The unit-circle construction is introduced as the tool for understanding the target graph of .
The ray and angle are drawn inside the newly created coordinate system and circle.
The unit-circle setup contains the standard-position angle representation used to locate a point on the circle.
The speaker links the ray intersecting the unit circle to the coordinates .
Once the angle is drawn, the unit-circle definition assigns the intersection point the coordinates .
The point label is followed by the equation .
The quotient identity is interpreted through the coordinates of the unit-circle point: sine is the -coordinate and cosine is the -coordinate.
The speaker says the y-coordinate over the x-coordinate is essentially the slope of this line.
is written alongside .
Within the unit-circle picture, the algebraic ratio and the geometric slope describe the same quantity.
After explaining slope, the speaker says, "So now let's make a table."
The slope interpretation motivates the next step: listing several angles and computing their tangent values for graphing.
The speaker says the chosen values will help us think about the shape of the graph of .
The table is built before any graph is drawn.
The geometric slope interpretation is converted into numerical table entries to prepare for graphing.
Specific rays on the unit circle are used to determine the values 1 and -1.
The ray on the unit circle provides the visual basis for concluding .
The speaker explicitly invokes sine of theta over cosine of theta to explain the value 1.
The identity is used to justify the special value at .
The presenter says these easy values will help form the shape of the graph.
The first tabulated point is (0,0).
The value is one of the sample points collected to infer the overall graph shape.
At 00:46.000 the speaker says, "let's just start plotting a few of these points" after completing the table.
The plotted points correspond exactly to the table entries.
The computed tangent values are applied to create the initial graph of .
At 01:12.000 the speaker contrasts the plotted points with the later claim that the graph is not a line, then discusses behavior near through 02:00.000.
The finite plotted points suggest simple linear behavior, but the later limiting argument shows an asymptote instead.
The speaker says he sets up a unit circle so we can visualize what the tangent of various thetas are.
The point is labeled .
The equation is written on screen.
The speaker says the ratio is essentially the slope of the line, change in y over change in x.
The speaker corrects an initial slip and states that the x-coordinate is cosine theta.
The speaker says, "So now let's make a table."
The speaker asks what is the slope of this ray when theta is zero radians and answers that it is zero.
The speaker explains that at the x-coordinate and y-coordinate are the same, so the slope is 1.
The table records -π/4 with tangent -1.
A fourth-quadrant ray is drawn for the negative angle.
The verbal explanation is incomplete at the end of the clip.
The speaker says he is picking values that will help us think about the shape of the graph of .
The speaker says tangent of theta is sine of theta over cosine of theta, and they are the same thing at this angle.
The full explanation of occurs from 00:00.000 to 00:45.000.
Covered · Opening statement of purpose and visible title .
Covered · Auxiliary axes and unit circle are drawn to visualize tangent values.
Covered · A ray from the origin defines angle in standard position.
Covered · The intersection point is labeled , with an audible self-correction.
Covered · The quotient identity is stated and written.
Covered · The ratio is reinterpreted as the slope of the terminal ray.
Covered · The diagram is simplified and a new table is begun, but no entries are completed within the clip.
Covered · Opening setup shows the title , the blank table, the unit circle, and the empty coordinate grid.
Covered · The presenter evaluates by noting the ray is horizontal and therefore has slope 0.
Covered · The speaker explains the strategy of choosing easy angles to infer the shape of the graph.
Covered · The angle is introduced, drawn on the unit circle, and justified as having tangent 1 through equal coordinates and equality.
Covered · The angle -π/4 is introduced and written in the table with tangent -1; the spoken explanation begins but the clip ends before it is fully developed.
Covered · Unit-circle coordinates for are explained and used to compute .
Covered · The unit circle is cleaned up and the table entry for is finalized as .
Covered · The graph axes are identified and the first three points of are plotted.
Covered · The speaker warns the graph is not a line, explains growth toward infinity near , and draws a vertical asymptote.
Covered · Initial layout and setup for the tangent graph are visible; the spoken content in this opening portion is fragmentary, but it leads directly into the slope-based explanation.
Covered · The speaker explains unbounded positive and negative slope behavior near ± while the central branch is drawn.
Covered · The clip explicitly identifies a vertical asymptote and completes the central branch between -π/2 and .
Covered · The instructor continues past , explaining why the next branch starts from very negative values and rises again.
Covered · A concrete example angle is introduced and rewritten as , though the example is not completed within the clip.
Covered · The entire supplied clip is a continuous whiteboard explanation of tangent values, unit-circle slopes, plotted points, asymptotes, and periodic repetition.
Covered · The entire clip focuses on introducing the shape and periodic nature of the tangent graph.
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