Introduction to the unit circle | Trigonometry | Khan Academy

1. Diagram
   Observation

   A horizontal axis is labeled x on the whiteboard.

x

horizontal coordinate axis in the Cartesian plane

real numbers

1. Diagram
   Observation

   A vertical axis is labeled y on the whiteboard.

y

vertical coordinate axis in the Cartesian plane

real numbers

1. Audio
   Observation

   The speaker says, "So this is a positive angle theta."

2. Diagram
   Observation

   A curved arrow from the positive x-axis toward the terminal ray is labeled θ.

$\theta$

a positive angle measured counterclockwise from the initial side along the positive x-axis

angle measure

1. Audio
   Observation

   The speaker says the intersection point has coordinates a comma b and that the x value where it intersects is a.

2. Diagram
   Observation

   The point on the unit circle is labeled (a,b).

a

x-coordinate of the intersection point between the terminal side of the angle and the unit circle

real number

1. Audio
   Observation

   The speaker says the intersection point has coordinates a comma b and that the y value where it intersects is b.

2. Diagram
   Observation

   The point on the unit circle is labeled (a,b).

b

y-coordinate of the intersection point between the terminal side of the angle and the unit circle

real number

1. Diagram
   Observation

   The angle at the origin between the positive x-axis and the radius to (a,b) is labeled θ.

2. Audio
   Observation

   The speaker says, "I want to make this theta part of a right triangle."

$\theta$

The angle formed at the origin by the positive x-axis and the radius to the point (a,b).

An acute angle in the constructed first-quadrant right triangle shown in this clip.

1. Diagram
   Observation

   The point on the circle is labeled (a,b), and the horizontal leg from the origin to the foot of the altitude is later labeled a.

2. Audio
   Observation

   The speaker says the base length is "of length a" and that it is the x-coordinate of the point of intersection.

a

The x-coordinate of the point where the terminal side meets the unit circle; also the adjacent-side length of the constructed right triangle.

In this clip, a is shown as a positive horizontal length in the first quadrant.

1. Diagram
   Observation

   The point on the circle is labeled (a,b), and the vertical leg is later labeled b.

2. Audio
   Observation

   The speaker says the height is equal to b and identifies b as the y-coordinate of the point of intersection.

b

The y-coordinate of the point where the terminal side meets the unit circle; also the opposite-side length of the constructed right triangle.

In this clip, b is shown as a positive vertical length in the first quadrant.

1. Audio
   Observation

   The speaker says, "this hypotenuse is just a radius of a unit circle. The unit circle has a radius of one, so the hypotenuse has length one."

2. Diagram
   Observation

   The slanted segment from the origin to (a,b) is marked with length 1.

1

The length of the hypotenuse of the constructed right triangle, because it is a radius of the unit circle.

A constant length in this clip.

1. Formula
   Observation

   The board shows $\cos \theta$ = and then the completed expression $\frac{a}{1}$.

2. Audio
   Observation

   The speaker asks for the cosine of the angle and uses the CAH part of SOH CAH TOA.

$\cos\theta$

The cosine of the angle θ, defined here by the right-triangle ratio adjacent over hypotenuse.

Applied to the acute angle θ in the constructed right triangle.

1. Diagram
   Observation

   Angle $\theta$ is drawn at the origin between the positive x-axis and the terminal side of the angle.

$\theta$

The angle formed by the terminal side and the positive x-axis.

Real numbers (angles).

1. Diagram
   Observation

   Point $(a,b)$ is on the unit circle; $a$ is the x-coordinate and the length of the adjacent side of the right triangle.

2. Formula
   Observation

   $\cos \theta = a$

a

x-coordinate of the point where the terminal side intersects the unit circle; length of the adjacent side.

Real numbers, $[-1, 1]$.

1. Audio
   Observation

   "What I have attempted to draw here is a unit circle." "And the fact that I'm calling it a unit circle means it has a radius of one." "So this length from the center, and I centered it at the origin, this length from the center to any point on the circle is of length one."

2. Diagram
   Observation

   A yellow circle centered at the origin is drawn on the x-y axes, and a yellow segment from the origin to the rightmost point of the circle is added to indicate the radius.

The video defines the drawn object as a unit circle centered at the origin, with radius equal to 1 from the center to any point on the circle.

1. centered at the origin

2. radius is 1

1. x
2. y

1. Audio
   Observation

   The speaker identifies the four intercepts in order: (1,0), (0,1), (-1,0), and (0,-1), explaining each by how far the point lies from the origin along the axes.

2. Diagram
   Observation

   The labels (1,0), (0,1), (-1,0), and (0,-1) are written at the corresponding points where the circle meets the axes.

Because the circle has radius 1 and is centered at the origin, its intersections with the coordinate axes are read directly as one unit right, up, left, and down from the origin.

1. the circle is centered at the origin

2. the radius is 1

1. Unit circle definition

1. Audio
   Observation

   "I'm going to define a convention for positive angles." "The initial side of an angle, we're always going to do along the positive x-axis." "To draw an angle, a positive angle, the terminal side, we're going to move in a counterclockwise direction." "You can imagine a negative angle would move in a clockwise direction."

2. Diagram
   Observation

   The text (+) -> counterclockwise and (-) -> clockwise is written above the circle.

The video sets the standard trigonometric convention: the initial side lies on the positive x-axis, positive angles rotate counterclockwise, and negative angles rotate clockwise.

1. initial side is along the positive x-axis

2. positive rotation is counterclockwise

3. negative rotation is clockwise

1. x

1. Audio
   Observation

   "This is the initial side, and then from that I go in a counterclockwise direction until I get until I measure out the angle and then this is the terminal side. So this is a positive angle theta."

2. Diagram
   Observation

   A pink ray is drawn from the origin into the first quadrant, and a curved arrow from the positive x-axis to that ray is labeled θ.

A positive angle is shown as starting from the initial side on the positive x-axis and ending at a terminal side reached by counterclockwise rotation; the example angle is labeled θ.

1. the angle is positive

2. rotation is counterclockwise from the positive x-axis

1. Convention for positive and negative angles
2. $\theta$

1. Audio
   Observation

   "And what I want to do is think about this point of intersection between the terminal side of this angle and my unit circle. And let's just say it has the coordinates a comma b. The x value where it intersects is a, the y value where it intersects is b."

2. Diagram
   Observation

   The intersection point on the terminal ray and the circle is marked and labeled (a,b).

The terminal side of the angle meets the unit circle at a point whose coordinates are introduced as (a,b), with a the x-coordinate and b the y-coordinate.

1. the point lies on the terminal side of the angle

2. the point lies on the unit circle

1. Unit circle definition
2. Initial side, terminal side, and positive angle θ
3. a
4. b

1. Audio
   Observation

   The speaker says the goal is to see how the unit circle can help extend traditional definitions of trig functions.

2. Diagram
   Observation

   A yellow unit circle centered at the origin is drawn with axes, key points (1,0), (0,1), (-1,0), (0,-1), and a point (a,b) on the circle.

3. Animation
   Observation

   From about 16s to 23s, a teal vertical segment is dropped from (a,b) to the x-axis and a right-angle mark is added.

The clip begins with a unit circle and a point (a,b) on it. The angle θ is taken from the positive x-axis to the radius ending at (a,b). To connect this picture to the familiar right-triangle definition of trigonometric functions, the speaker drops a perpendicular from (a,b) to the x-axis, creating a right triangle whose vertices are the origin, (a,0), and (a,b).

1. The construction is shown for a first-quadrant point (a,b) on the unit circle.

2. The perpendicular is dropped to the x-axis to form a right triangle containing θ.

1. Audio
   Observation

   The speaker asks, "what is the length of the hypotenuse?" and answers that it is a radius of a unit circle, hence length one.

2. Diagram
   Observation

   The slanted side from the origin to (a,b) is identified as the hypotenuse and marked 1.

In the right triangle formed inside the unit circle, the hypotenuse is exactly the radius from the origin to the point (a,b). Because the circle is a unit circle, that radius has length 1, so the hypotenuse length is 1.

1. The triangle is built from the origin to a point on the unit circle.

2. The circle has radius 1.

1. Using the unit circle to build a right triangle for θ

1. Audio
   Observation

   The speaker says the blue side can be viewed as the opposite side to the angle and that its height is the same thing as the y-coordinate of the point of intersection.

2. Diagram
   Observation

   The vertical leg is labeled b, and a dashed horizontal guide connects (a,b) to the y-axis at height b.

The vertical leg of the right triangle is the side opposite θ. Its length matches the y-coordinate of the point (a,b), so that opposite side has length b.

1. The point on the circle is (a,b).

2. The triangle is formed by dropping a perpendicular from (a,b) to the x-axis.

1. Using the unit circle to build a right triangle for θ

1. Audio
   Observation

   The speaker says the base is the x-coordinate of the point of intersection and that the length from the origin to that foot is a.

2. Diagram
   Observation

   The horizontal leg from the origin to (a,0) is labeled a.

The horizontal leg of the right triangle lies along the x-axis from the origin to the foot of the perpendicular. Its length is the x-coordinate of the point (a,b), so the adjacent side has length a.

1. The point on the circle is (a,b).

2. The triangle is formed by dropping a perpendicular from (a,b) to the x-axis.

1. Using the unit circle to build a right triangle for θ

1. Audio
   Observation

   The speaker says, "we just need our SOH CAH TOA definition" and explains that the CAH part tells us cosine equals adjacent over hypotenuse.

2. Diagram
   Observation

   The board writes SOH CAH TOA, and CAH is boxed.

The clip explicitly invokes the mnemonic SOH CAH TOA. The CAH portion states that the cosine of an angle in a right triangle is the adjacent side divided by the hypotenuse.

1. This is the traditional right-triangle definition being used before extension to the full unit circle.

2. It applies to the acute angle θ in the constructed right triangle.

1. Hypotenuse of the constructed triangle equals the unit-circle radius
2. Adjacent side length equals the x-coordinate a

1. Audio
   Observation

   Speaker states cosine is adjacent over hypotenuse, and since hypotenuse is 1, cosine equals a.

2. Formula
   Observation

   $\cos \theta = \frac{a}{1} = a$

On the unit circle, the cosine of an angle $\theta$ is defined as the x-coordinate ($a$) of the point where the terminal side of the angle intersects the circle. This is derived from the SOH CAH TOA definition where cosine is adjacent over hypotenuse, and the hypotenuse (radius) is 1.

1. The angle is in standard position on the unit circle.

1. Audio
   Observation

   Speaker states sine is opposite over hypotenuse, and since hypotenuse is 1, sine equals b.

2. Formula
   Observation

   $\sin \theta = \frac{b}{1} = b$

On the unit circle, the sine of an angle $\theta$ is defined as the y-coordinate ($b$) of the point where the terminal side of the angle intersects the circle. This is derived from the SOH CAH TOA definition where sine is opposite over hypotenuse, and the hypotenuse (radius) is 1.

1. The angle is in standard position on the unit circle.

1. Audio
   Observation

   The speaker asks what the cosine is in terms of a and b, then applies CAH and says the adjacent side has length a.

2. Formula
   Observation

   The board shows $\cos \theta$ = followed by $\frac{a}{1}$.

1. The clip ends immediately after writing $a/1$; it does not continue to simplify to a within this segment.

In the right triangle formed from the unit circle point (a,b), the cosine of θ is the adjacent side length a divided by the hypotenuse length 1, so the displayed expression is $\cos\theta=\frac{a}{1}$.

1. A point (a,b) lies on the unit circle.

2. A perpendicular is dropped from (a,b) to the x-axis to form a right triangle containing θ.

3. The hypotenuse is the radius of the unit circle and therefore has length 1.

4. The adjacent side is the horizontal leg of length a.

For the specific angle θ and point (a,b) shown in this clip.

1. Audio
   Observation

   Speaker states the point $(a,b)$ can be viewed as $(\cos \theta, \sin \theta)$.

2. Formula
   Observation

   (a, b) = ($\cos \theta$, $\sin \theta$)

For any angle $\theta$ in standard position, the coordinates of the point where its terminal side intersects the unit circle are $(\cos \theta, \sin \theta)$.

1. The angle is in standard position.

2. The circle is a unit circle (radius 1).

For all angles $\theta$.

1. Audio
   Observation

   For each axis intersection, the speaker explains the coordinate by saying how far the point is from the origin horizontally and vertically.

2. Diagram
   Observation

   The four labeled points lie exactly where the circle crosses the axes.

1. Expression

   $$r = 1$$
   Explanation

   Start from the stated property of the unit circle: every point on it is distance 1 from the origin.

   Justification

   Given in the audio as the meaning of “unit circle.”

   Shown in the video
2. Expression

   $$(1,0)$$
   Explanation

   On the positive x-axis, the point is 1 unit to the right of the origin and 0 units vertically, so its coordinates are (1,0).

   Justification

   Direct reading of horizontal and vertical displacement from the origin.

   Shown in the video
3. Expression

   $$(0,1)$$
   Explanation

   On the positive y-axis, the point is 1 unit above the origin and 0 units horizontally, so its coordinates are (0,1).

   Justification

   Direct reading of horizontal and vertical displacement from the origin.

   Shown in the video
4. Expression

   $$(-1,0)$$
   Explanation

   On the negative x-axis, the point is 1 unit to the left of the origin and 0 units vertically, so its coordinates are (-1,0).

   Justification

   Direct reading of horizontal and vertical displacement from the origin.

   Shown in the video
5. Expression

   $$(0,-1)$$
   Explanation

   On the negative y-axis, the point is 1 unit below the origin and 0 units horizontally, so its coordinates are (0,-1).

   Justification

   Direct reading of horizontal and vertical displacement from the origin.

   Shown in the video

The unit circle centered at the origin intersects the coordinate axes at (1,0), (0,1), (-1,0), and (0,-1).

1. Audio
   Observation

   The speaker determines the hypotenuse, opposite side, and adjacent side, then applies SOH CAH TOA to compute cosine.

2. Formula
   Observation

   The final written result is $\cos \theta = \frac{a}{1}$.

1. Expression

   $$\text{Construct a right triangle from } (0,0), (a,0), (a,b)$$
   Explanation

   Drop a perpendicular from the point (a,b) on the unit circle to the x-axis so that θ becomes an angle of a right triangle.

   Justification

   Explicit construction shown and described in the video.

   Shown in the video
2. Expression

   $$\text{hypotenuse}=1$$
   Explanation

   The slanted side from the origin to (a,b) is a radius of the unit circle, and the unit circle has radius 1.

   Justification

   Stated directly by the speaker and marked on the diagram.

   Shown in the video
3. Expression

   $$\text{opposite}=b$$
   Explanation

   The vertical leg equals the y-coordinate of the point (a,b).

   Justification

   Stated directly by the speaker and labeled on the diagram.

   Shown in the video
4. Expression

   $$\text{adjacent}=a$$
   Explanation

   The horizontal leg equals the x-coordinate of the point (a,b).

   Justification

   Stated directly by the speaker and labeled on the diagram.

   Shown in the video
5. Expression

   $$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$$
   Explanation

   Use the CAH part of SOH CAH TOA for the right-triangle definition of cosine.

   Justification

   Explicitly invoked in the audio and written on the board.

   Shown in the video
6. Expression

   $$\cos\theta=\frac{a}{1}$$
   Explanation

   Substitute adjacent = a and hypotenuse = 1 into the CAH formula.

   Justification

   Direct substitution from the previous steps.

   Shown in the video

The clip derives the displayed identity $\cos\theta=\frac{a}{1}$ for the constructed right triangle inside the unit circle.

1. Audio
   Observation

   Step-by-step derivation using SOH CAH TOA and the fact that the hypotenuse is 1.

2. Formula
   Observation

   $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{1} = a$; $\quad \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{1} = b$

1. Expression

   $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$
   Explanation

   Start with the CAH part of SOH CAH TOA.

   Justification

   Definition of cosine in a right triangle.

   Shown in the video
2. Expression

   $$\cos \theta = \frac{a}{1}$$
   Explanation

   Substitute the adjacent side length $a$ and hypotenuse length 1 (radius of unit circle).

   Justification

   Visual identification of sides in the diagram.

   Shown in the video
3. Expression

   $$\cos \theta = a$$
   Explanation

   Simplify the fraction.

   Justification

   Arithmetic simplification.

   Shown in the video
4. Expression

   $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$
   Explanation

   Start with the SOH part of SOH CAH TOA.

   Justification

   Definition of sine in a right triangle.

   Shown in the video
5. Expression

   $$\sin \theta = \frac{b}{1}$$
   Explanation

   Substitute the opposite side length $b$ and hypotenuse length 1.

   Justification

   Visual identification of sides in the diagram.

   Shown in the video
6. Expression

   $$\sin \theta = b$$
   Explanation

   Simplify the fraction.

   Justification

   Arithmetic simplification.

   Shown in the video

On the unit circle, $\cos \theta$ equals the x-coordinate $a$, and $\sin \theta$ equals the y-coordinate $b$.

1. Audio
   Observation

   Speaker transitions from SOH CAH TOA definitions to unit circle definitions.

2. Formula
   Observation

   Visual comparison between SOH CAH TOA formulas and new unit circle definitions.

1. Expression

   $$\cos \theta = \frac{a}{1} = a$$
   Explanation

   From SOH CAH TOA, cosine is adjacent over hypotenuse. In the unit circle, the hypotenuse is always 1, so cosine equals the adjacent side length a.

   Justification

   SOH CAH TOA definition combined with unit circle radius property

   Shown in the video
2. Expression

   $$\sin \theta = \frac{b}{1} = b$$
   Explanation

   From SOH CAH TOA, sine is opposite over hypotenuse. In the unit circle, the hypotenuse is always 1, so sine equals the opposite side length b.

   Justification

   SOH CAH TOA definition combined with unit circle radius property

   Shown in the video
3. Expression

   $$\cos \theta = x$$
   Explanation

   Since the adjacent side length a corresponds to the x-coordinate of the intersection point, cosine is defined as the x-coordinate.

   Justification

   Geometric correspondence between triangle legs and coordinates

   Shown in the video
4. Expression

   $$\sin \theta = y$$
   Explanation

   Since the opposite side length b corresponds to the y-coordinate of the intersection point, sine is defined as the y-coordinate.

   Justification

   Geometric correspondence between triangle legs and coordinates

   Shown in the video
5. Expression

   $$\tan \theta = \frac{y}{x}$$
   Explanation

   Tangent is defined as sine divided by cosine, which becomes y divided by x in the unit circle context.

   Justification

   Definition of tangent as ratio of sine to cosine

   Shown in the video

Trigonometric functions can be generalized from right triangle ratios to coordinate-based definitions using the unit circle.

1. Audio
   Observation

   The speaker draws a positive angle, names it theta, and then discusses the point where its terminal side intersects the unit circle as (a,b).

2. Diagram
   Observation

   A ray from the origin into the first quadrant is drawn, a counterclockwise arc labeled θ is added, and the intersection point is labeled (a,b).

Draw a positive angle in standard position on the unit circle and identify the coordinates of the point where its terminal side meets the circle.

1. unit circle centered at the origin

2. initial side along the positive x-axis

3. positive angles rotate counterclockwise

Produce a concrete example of a positive angle θ and name the intersection point (a,b).

1. Expression

   $$\text{initial side on } +x\text{-axis}$$
   Explanation

   Begin with the initial side placed along the positive x-axis.

   Justification

   This is the convention stated in the video for angles in this setup.

   Shown in the video
2. Expression

   $$\text{rotate counterclockwise}$$
   Explanation

   Move from the initial side in the counterclockwise direction to form a positive angle.

   Justification

   The video explicitly defines positive angles this way.

   Shown in the video
3. Expression

   $$\theta$$
   Explanation

   Label the resulting angle θ.

   Justification

   The speaker names the drawn example as a positive angle theta.

   Shown in the video
4. Expression

   $$(a,b)$$
   Explanation

   Mark the point where the terminal side intersects the unit circle and call its coordinates (a,b).

   Justification

   The speaker introduces this notation directly in the audio and writes it on the diagram.

   Shown in the video

A positive angle θ is drawn counterclockwise from the positive x-axis, and its terminal side meets the unit circle at the point (a,b).

The construction matches the stated convention: initial side on the positive x-axis, counterclockwise rotation for a positive angle, and the labeled intersection point on the circle.

1. Diagram
   Observation

   A unit circle with point (a,b), angle θ, and a constructed right triangle is used throughout.

2. Audio
   Observation

   The speaker works through the side lengths and then computes cosine from them.

1. No numerical coordinates are assigned to (a,b); the example remains symbolic.

Given a point (a,b) on the unit circle and the angle θ from the positive x-axis to the radius ending at (a,b), construct the associated right triangle and express cos θ in terms of a and b.

1. A unit circle centered at the origin.

2. A point (a,b) on the circle.

3. The angle θ at the origin between the positive x-axis and the radius to (a,b).

4. A perpendicular dropped from (a,b) to the x-axis.

Find the right-triangle expression for cos θ.

1. Expression

   $$\text{Form triangle with vertices }(0,0),(a,0),(a,b)$$
   Explanation

   Drop the altitude from (a,b) to the x-axis to make θ part of a right triangle.

   Justification

   Explicit construction in the video.

   Shown in the video
2. Expression

   $$\text{hypotenuse}=1$$
   Explanation

   The radius of the unit circle is the hypotenuse.

   Justification

   Stated in the video.

   Shown in the video
3. Expression

   $$\text{adjacent}=a$$
   Explanation

   The horizontal leg is the x-coordinate of (a,b).

   Justification

   Stated in the video.

   Shown in the video
4. Expression

   $$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$$
   Explanation

   Apply the CAH rule from SOH CAH TOA.

   Justification

   Stated in the video.

   Shown in the video
5. Expression

   $$\cos\theta=\frac{a}{1}$$
   Explanation

   Substitute the identified side lengths into the cosine ratio.

   Justification

   Direct substitution.

   Shown in the video

$\cos\theta=\frac{a}{1}$

The result follows from the labeled side lengths on the diagram and the explicitly stated CAH definition.

1. Diagram
   Observation

   A yellow circle is shown centered at the origin on black background, with x and y axes crossing at the center; a yellow radius segment is drawn from the origin to the right side of the circle.

2. Audio
   Observation

   The speaker states that the figure is a unit circle and that its radius is one.

1. yellow circle

2. x-axis

3. y-axis

4. origin

5. yellow radius segment

1. The radius segment is added from the center to the circle after the circle is already visible.

1. The circle remains centered at the origin.

2. The axes remain fixed as the reference frame.

The visual emphasizes that the defining feature of the unit circle is distance 1 from the center to any point on the circle.

1. Diagram
   Observation

   The labels (1,0), (0,1), (-1,0), and (0,-1) appear sequentially at the four axis intersections of the circle.

2. Audio
   Observation

   The speaker asks what each coordinate would be and answers each one in turn.

1. circle

2. coordinate axes

3. four labeled points

1. Point labels are added one by one around the circle in the order right, top, left, bottom.

1. The circle and axes do not move.

2. Each labeled point stays on both the circle and an axis.

The sequence shows how radius 1 translates directly into the four basic coordinate points on the unit circle.

1. Diagram
   Observation

   Above the circle, the text (+) -> counterclockwise and (-) -> clockwise is written.

2. Audio
   Observation

   The speaker explains the convention for positive and negative angles.

1. text annotation (+) -> counterclockwise

2. text annotation (-) -> clockwise

1. The convention text is added above the existing unit-circle diagram.

1. The underlying circle and axes remain unchanged.

The annotation establishes the rule used for all later angle drawings in the clip.

1. Diagram
   Observation

   A pink ray is drawn from the origin into the first quadrant, a curved arrow from the positive x-axis to the ray is labeled θ, and the intersection point on the circle is labeled (a,b).

2. Audio
   Observation

   The speaker describes the initial side, terminal side, positive angle theta, and the intersection coordinates a comma b.

1. initial side on positive x-axis

2. pink terminal ray

3. curved angle arrow labeled θ

4. point labeled (a,b)

1. The terminal ray is added.

2. The counterclockwise angle marker labeled θ is added.

3. The intersection point label (a,b) is added.

1. The initial side remains on the positive x-axis.

2. The point (a,b) remains on both the terminal ray and the unit circle.

This visual links an abstract angle to a specific point on the unit circle, setting up the later connection between angles and coordinates.

1. Animation
   Observation

   A teal vertical segment is drawn downward from (a,b) to the x-axis, and a right-angle square is added at the foot.

2. Audio
   Observation

   The speaker says, "let me drop an altitude right over here" and notes the 90-degree angle.

1. Unit circle centered at the origin

2. Point (a,b) on the circle

3. Radius from origin to (a,b)

4. Angle θ at the origin

5. Teal vertical segment from (a,b) to the x-axis

6. Right-angle marker at (a,0)

1. A new vertical segment appears from (a,b) down to the x-axis.

2. A right-angle marker is added at the foot of that segment.

3. The picture changes from a circle-and-radius diagram to a right triangle embedded in the unit circle.

1. The circle remains centered at the origin.

2. The point (a,b) stays fixed on the circle.

3. The radius from the origin to (a,b) remains the same segment.

The animation shows how to convert the unit-circle picture into a right triangle so that the traditional trigonometric ratio definitions can be applied to θ.

1. Animation
   Observation

   The hypotenuse is marked 1, the vertical leg is labeled b, and the horizontal leg is labeled a.

2. Audio
   Observation

   The speaker identifies each side length in turn.

1. Hypotenuse from origin to (a,b)

2. Vertical leg

3. Horizontal leg

4. Labels 1, b, a

1. The slanted radius is labeled 1.

2. The vertical leg is labeled b.

3. The horizontal leg is labeled a.

4. A dashed horizontal guide from (a,b) to the y-axis visually links b to the y-coordinate.

1. The triangle shape and vertex positions do not change.

2. The point remains (a,b) on the unit circle.

The labeling establishes the exact correspondence between triangle side lengths and coordinates: hypotenuse = 1, opposite = b, adjacent = a.

1. Animation
   Observation

   The board writes cos θ =, then SOH CAH TOA with CAH boxed, and finally $\frac{a}{1}$.

2. Audio
   Observation

   The speaker asks for cosine and explains the CAH rule before substituting a and 1.

1. Expression cos θ =

2. Mnemonic SOH CAH TOA

3. Box around CAH

4. Fraction $\frac{a}{1}$

1. The prompt cos θ = is written.

2. SOH CAH TOA is written and CAH is emphasized by boxing.

3. The final fraction $\frac{a}{1}$ is written as the value of cos θ.

1. The underlying triangle and labels a, b, 1 remain unchanged.

The visual sequence shows the transition from geometric labels to the algebraic right-triangle definition of cosine.

1. Animation
   Observation

   Speaker draws several right triangles with increasing height to show the angle approaching 90 degrees but never reaching it.

1. Right triangles

1. Height of the triangle increases

2. Angle approaches 90 degrees

1. Base remains horizontal

2. One angle remains 90 degrees

Demonstrates why SOH CAH TOA cannot handle angles of 90 degrees or more, as a second 90-degree angle would be required, which is impossible in a Euclidean triangle.

1. Diagram
   Observation

   Complete visual layout showing unit circle, coordinate axes, right triangle, and all mathematical notations.

1. yellow unit circle

2. black coordinate axes

3. purple angle arc

4. cyan right triangle

5. green adjacent side

6. magenta opposite side

7. white hypotenuse

1. Text annotations appear sequentially as speaker defines each function

2. Ditto marks used to avoid repeating full phrases

1. Circle remains centered at origin with radius 1

2. Triangle maintains right angle at x-axis

3. Angle θ remains at origin

The diagram visually demonstrates how trigonometric functions correspond to coordinates on the unit circle, with the right triangle providing the bridge between SOH CAH TOA and coordinate definitions.

1. Audio
   Observation

   The speaker explicitly contrasts positive angles moving counterclockwise with negative angles moving clockwise.

2. Diagram
   Observation

   The written convention (+) -> counterclockwise and (-) -> clockwise appears on screen.

One might assume that any rotation direction can be called positive without a fixed convention.

In this lesson, positive angles are defined to rotate counterclockwise from the positive x-axis, while negative angles rotate clockwise.

1. Audio
   Observation

   The speaker says the point of the exercise is to see how the unit circle can help extend traditional definitions of trig functions, and later says, "We are actually in the process of extending it."

One may think trigonometric functions are only defined by right-triangle side ratios.

The clip explicitly presents SOH CAH TOA as the current traditional definition and uses the unit circle as the device for extending that definition beyond the original right-triangle setting.

1. Audio
   Observation

   Speaker explicitly states 'SOH CAH TOA has a problem... it starts to break down' for angles outside $(0, 90^\circ)$.

Students often believe the right-triangle definitions of sine and cosine apply to any angle measure.

SOH CAH TOA is strictly limited to acute angles ($0^\circ < \theta < 90^\circ$) because it requires a right triangle. The unit circle definition extends these concepts to all real angles.

1. Audio
   Observation

   Speaker begins by reciting SOH CAH TOA then immediately proposes unit circle alternative.

2. Formula
   Observation

   SOH CAH TOA formulas shown separately from unit circle definitions.

Students may believe SOH CAH TOA is the only way to define trigonometric functions.

SOH CAH TOA only works for acute angles in right triangles, while the unit circle definition extends trigonometric functions to all real angles.

1. Audio
   Observation

   The speaker uses the radius-1 property of the unit circle to determine the coordinates where the circle meets the axes.

The definition of the unit circle as radius 1 centered at the origin is applied to read off the four axis-intercept coordinates.

1. Audio
   Observation

   After stating the convention, the speaker immediately draws a positive angle and labels it theta.

2. Diagram
   Observation

   The drawn angle follows the previously written counterclockwise rule.

The general convention for positive angles is instantiated in the specific example angle θ.

1. Audio
   Observation

   The speaker turns attention from the angle itself to the point where the terminal side intersects the unit circle and names that point (a,b).

2. Diagram
   Observation

   The label (a,b) is placed exactly at the intersection of the terminal ray and the circle.

The terminal side of the angle θ determines a unique point on the unit circle, which is then described by coordinates (a,b).

1. Audio
   Observation

   The speaker builds a right triangle from the unit circle and then computes cosine from its side lengths.

2. Diagram
   Observation

   The unit-circle point (a,b) supplies the side labels a, b, and 1 used in the cosine formula.

The unit-circle construction provides the right triangle whose side lengths are then used to apply the right-triangle definition of cosine.

1. Audio
   Observation

   The speaker says the CAH part tells us cosine equals adjacent over hypotenuse.

2. Formula
   Observation

   The board boxes CAH and then writes $\frac{a}{1}$ for cos θ.

The claim $\cos θ = a/1$ depends directly on the CAH rule from SOH CAH TOA.

1. Audio
   Observation

   The speaker identifies the base as the x-coordinate of the point of intersection and labels it a.

2. Diagram
   Observation

   The horizontal leg is labeled a beneath the point (a,b).

The numerator a in $\cos θ = a/1$ comes from identifying the adjacent side with the x-coordinate of the unit-circle point.

1. Audio
   Observation

   The speaker says the hypotenuse is a radius of the unit circle and therefore has length one.

2. Diagram
   Observation

   The slanted side is labeled 1.

The denominator 1 in $\cos θ = a/1$ comes from recognizing the hypotenuse as the unit-circle radius.

1. Audio
   Observation

   Speaker says 'Let's set up a new definition... which is really an extension of SOH CAH TOA'.

The unit circle definitions of sine and cosine generalize the SOH CAH TOA definitions, allowing them to work for angles beyond the acute range.

1. Audio
   Observation

   Speaker explicitly states using unit circle to define trig functions.

2. Diagram
   Observation

   Visual connection between circle geometry and function definitions.

The unit circle provides the geometric framework that allows cosine to be defined as a coordinate value rather than a triangle ratio.

1. Audio
   Observation

   Speaker derives tangent from sine and cosine definitions.

2. Formula
   Observation

   Written equation shows $\tan θ = \sin θ / \cos θ$.

The definition of tangent depends directly on the prior definitions of sine and cosine as coordinate values.

1. Audio
   Observation

   The speaker defines the figure as a unit circle with radius one centered at the origin.

1. Unit circle definition

1. Audio
   Observation

   The speaker lists the coordinates where the circle intersects the axes.

1. Coordinates of the unit circle's axis intersections
2. Deriving the four axis-intercept coordinates from radius 1

1. Audio
   Observation

   The speaker states the convention for positive and negative angle directions.

1. Convention for positive and negative angles

1. Audio
   Observation

   The speaker distinguishes the initial side from the terminal side while drawing θ.

1. Initial side, terminal side, and positive angle θ

1. Audio
   Observation

   The speaker introduces (a,b) as the coordinates of the intersection point between the terminal side and the unit circle.

1. Intersection point of the terminal side with the unit circle
2. a
3. b

1. Audio
   Observation

   The speaker says he wants to make theta part of a right triangle and drops an altitude.

2. Animation
   Observation

   A perpendicular is drawn from (a,b) to the x-axis.

1. Using the unit circle to build a right triangle for θ

1. Audio
   Observation

   The speaker says the hypotenuse is a radius of a unit circle and the radius is one.

1. Hypotenuse of the constructed triangle equals the unit-circle radius

1. Audio
   Observation

   The speaker calls the vertical side the opposite side and equates it to the y-coordinate b.

1. Opposite side length equals the y-coordinate b

1. Audio
   Observation

   The speaker identifies the base as the x-coordinate and labels it a.

1. Adjacent side length equals the x-coordinate a

1. Audio
   Observation

   The speaker applies the CAH part of SOH CAH TOA and substitutes adjacent = a and hypotenuse = 1.

2. Formula
   Observation

   The board shows $\cos θ = a/1$.

1. CAH rule from SOH CAH TOA
2. For the constructed triangle, $\cos θ = a/1$
3. Deriving cos θ from the unit-circle right triangle

1. Audio
   Observation

   Explanation of why cosine equals the x-coordinate on the unit circle.

1. Cosine Definition on Unit Circle
2. Derivation of Sine and Cosine from SOH CAH TOA on Unit Circle

1. Audio
   Observation

   Discussion on the breakdown of SOH CAH TOA for large or negative angles.

1. Limitations of SOH CAH TOA
2. Misconception: SOH CAH TOA works for all angles