The Law of Sines

1. Audio
   Observation

   The speaker says the law applies to an oblique triangle with angles A, B, and C, all capitalized.

2. Formula
   Observation

   The displayed formula uses $\sin A$, $\sin B$, and $\sin C$ in denominators.

3. Diagram
   Observation

   The triangle diagram labels vertices A, B, and C.

A

Angle at vertex A of the triangle; in the worked example it is the angle found from the angle-sum relation.

Interior angle of a triangle, measured in degrees in the worked example.

1. Audio
   Observation

   The speaker names angles A, B, and C.

2. Formula
   Observation

   The displayed formula contains $\sin B$.

3. Diagram
   Observation

   Vertex B is labeled on the triangle.

B

Angle at vertex B of the triangle; in the worked example it is given as 50°.

Interior angle of a triangle, measured in degrees in the worked example.

1. Audio
   Observation

   The speaker names angles A, B, and C.

2. Formula
   Observation

   The displayed formula contains $\sin C$.

3. Diagram
   Observation

   Vertex C is labeled on the triangle.

C

Angle at vertex C of the triangle; in the worked example it is given as 85°.

Interior angle of a triangle, measured in degrees in the worked example.

1. Audio
   Observation

   The speaker says sides a, b, and c are lowercase and precisely opposite their corresponding angles.

2. Formula
   Observation

   The displayed formula uses side a over $\sin A$.

3. Diagram
   Observation

   Side a is drawn opposite vertex A.

a

Side length opposite angle A.

Positive real length.

1. Audio
   Observation

   The speaker says sides a, b, and c are lowercase and precisely opposite their corresponding angles.

2. Formula
   Observation

   The displayed formula uses side b over $\sin B$.

3. Diagram
   Observation

   Side b is drawn opposite vertex B.

b

Side length opposite angle B; in the worked example its value is 7.

Positive real length.

1. Audio
   Observation

   The speaker says sides a, b, and c are lowercase and precisely opposite their corresponding angles.

2. Formula
   Observation

   The displayed formula uses side c over $\sin C$.

3. Diagram
   Observation

   Side c is drawn opposite vertex C.

c

Side length opposite angle C.

Positive real length.

1. Formula
   Observation

   The displayed law of sines formula is written with $\sin A$, $\sin B$, and $\sin C$.

2. Audio
   Observation

   The speaker reads the relationship as a over sine A equals b over sine B equals c over sine C.

$\sin$

Sine function applied to each interior angle of the triangle.

In this lesson the angles are shown in degree measure.

1. Diagram
   Observation

   In the SAA example, side a is labeled opposite the 45° angle.

2. Formula
   Observation

   The displayed ratio uses $a / \sin 45$°.

a

side length opposite the 45° angle in the SAA example

positive real number

1. Diagram
   Observation

   In the SAA example, side c is labeled opposite the 85° angle.

2. Formula
   Observation

   The displayed ratio uses $c / \sin 85$°.

c

side length opposite the 85° angle in the SAA example

positive real number

1. Diagram
   Observation

   The known side in the SAA triangle is labeled 7.

2. Formula
   Observation

   The middle fraction is $7 / \sin 50$°.

7

known side length opposite the 50° angle in the SAA example

positive real number

1. Diagram
   Observation

   The ASA triangle initially labels the unknown right-hand angle as B.

2. Formula
   Observation

   The displayed ratio contains $12 / \sin B$ before B is computed.

B

unknown angle in the ASA example, later evaluated as 50°

angle measure in degrees

1. Diagram
   Observation

   Side a is labeled opposite the 35° angle.

2. Formula
   Observation

   The displayed ratio uses $a / \sin 35$°.

a

side length opposite the 35° angle in the ASA example

positive real number

1. Audio
   Observation

   The speaker says, "Oblique triangles are ones that don't have a right angle."

2. Diagram
   Observation

   A triangle labeled "Oblique Triangle" appears with the parenthetical text "(no right angle)".

An oblique triangle is a triangle without a right angle. The video contrasts this with right triangles, for which earlier trigonometric relationships such as SOHCAHTOA were learned.

1. Applies to triangles in plane geometry.

2. The defining restriction is that the triangle has no 90° angle.

1. Audio
   Observation

   The speaker states the law for an oblique triangle with angles A, B, C and sides a, b, c opposite those angles.

2. Formula
   Observation

   The screen displays $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.

3. Diagram
   Observation

   The triangle diagram labels vertices A, B, C and opposite sides a, b, c.

For an oblique triangle, the ratio of each side length to the sine of its opposite angle is the same for all three side-angle pairs. The video presents this as the relationship used to find missing sides and angles when enough values are known.

1. The triangle is an oblique triangle.

2. Angles A, B, C are the triangle’s interior angles.

3. Sides a, b, c are opposite A, B, C respectively.

4. The sine values in the denominators must be nonzero for the displayed ratios to be meaningful.

1. Oblique triangle
2. $\sin$

1. Audio
   Observation

   The speaker says, "We just have to know a few of these values, and we will be able to solve for the rest."

2. Caption evidence
   Observation

   On-screen text reads "if we know some of these we can solve for the others".

The video explains that the Law of Sines is useful because knowing some of the side-angle quantities allows the remaining unknown quantities to be solved from the equal ratios.

1. At least enough information must be known to form a usable equation.

2. The unknown should be isolated algebraically from the ratio equality.

1. Law of Sines

1. Audio
   Observation

   The speaker says, "Let's look at this side angle angle triangle. That means we know one side and two of the angles."

2. Caption evidence
   Observation

   The section title reads "Using The Law of Sines for SAA Triangles".

3. Diagram
   Observation

   The worked triangle shows one known side and two known angles.

An SAA triangle is a triangle in which one side and two angles are known. In the video’s example, the known side is paired with one of the known angles after the third angle is computed.

1. One side length is given.

2. Two interior angles are given.

1. Oblique triangle

1. Audio
   Observation

   The speaker says, "First, given two angles, we can always get the third. We just subtract these two angles from 180."

2. Formula
   Observation

   The screen displays $A = 180 - (85 + 50)$, then $A = 45$.

When two angles of a triangle are known, the third angle is found by subtracting their sum from 180. In the example, the known angles 85° and 50° give the missing angle $A = 45$°.

1. The figure is a triangle.

2. Two interior angles are already known.

3. Angle measure is in degrees in the displayed calculation.

1. Audio
   Observation

   The speaker says that once there is one angle-side pair, its ratio can be used to get the other two sides, and describes bringing the angle up and evaluating.

2. Formula
   Observation

   The screen shows $a = 7(\sin 45^\circ / \sin 50^\circ)$ and $c = 7(\sin 85^\circ / \sin 50^\circ)$.

3. Diagram
   Observation

   The known pair is highlighted as side 7 opposite 50°.

1. The final numerical values of a and c are not shown within this clip.

After forming a known angle-side pair, the Law of Sines is rearranged so the unknown side equals the known side multiplied by the ratio of the sine of the unknown side’s opposite angle to the sine of the known angle. In the example, side 7 opposite 50° is used to express both a and c.

1. A complete angle-side pair must be known.

2. The target side must be opposite a known angle.

3. The sine of the known angle in the denominator must be nonzero.

1. Law of Sines
2. Finding the third angle from two known angles

1. Formula
   Observation

   The screen shows $a/\sin 45$° = $7/\sin 50$° = $c/\sin 85$° for the SAA triangle.

2. Audio
   Observation

   The narrator says to plug values into the law of sines and use a calculator to get approximate side lengths.

The video applies the Law of Sines by equating each side length to the sine of its opposite angle. In the SAA example, the known pair is side 7 opposite 50°, so the other unknown sides are solved from the same common ratio.

1. Angles are measured in degrees in this segment.

2. Each side must be paired with the angle opposite it.

1. Audio
   Observation

   The narrator says, 'When we use this technique, we will usually measure angles in degrees and just plug things into a calculator to get approximate side lengths.'

2. Formula
   Observation

   The bottom line first shows $a = 7$(sin 45°/sin 50°) and $c = 7$(sin 85°/sin 50°), then updates to $a = 6.46$ and $c = 9.10$.

For a side-angle-angle triangle, the video first identifies the known side-angle pair, then isolates each unknown side by multiplying the known side by the ratio of sines. The results are decimal approximations rather than exact forms.

1. Two angles and one side are known.

2. The known side is not required to lie between the known angles.

1. Law of Sines ratio setup

1. Audio
   Observation

   The narrator says, 'This is where the side we know is specifically the one in between the two known angles.'

2. Diagram
   Observation

   The ASA slide title reads 'Using The Law of Sines for ASA Triangles' and the known side 12 lies between the 35° and 95° angles.

The video defines angle-side-angle as the situation in which the known side is the side included between the two known angles. This distinguishes ASA from the earlier SAA setup.

1. Two angles are known.

2. The known side is the side between those two angles.

1. Audio
   Observation

   The narrator says, 'we will again subtract the known angles from 180 to get the third.'

2. Formula
   Observation

   The ASA slide shows $B = 180 - (95 + 35)$, then $B = 50$.

Before applying the Law of Sines in the ASA example, the video computes the missing angle by subtracting the sum of the two known angles from 180°. Once that third angle is known, all three side-angle pairs can be matched for the ratio.

1. The figure is a Euclidean triangle.

2. Two angle measures are already known.

1. Definition of an ASA triangle case

1. Audio
   Observation

   The narrator says, 'Here we have to first find the angle opposite this side length.'

2. Formula
   Observation

   The SSA slide shows $A = \sin ^{-1}$(4 sin 52° / 9) = 20.5°.

In the side-side-angle example, the video first uses the Law of Sines to find the angle opposite one of the known sides. The displayed computation takes the inverse sine of (4 sin 52°)/9 and reports 20.5°.

1. Two sides and one angle are known.

2. The chosen angle is opposite one of the known sides.

1. Law of Sines ratio setup

1. Audio
   Observation

   The narrator says, 'Once we have that, we can subtract these two angles from 180 to get the third, and find the third side length if we need to.'

2. Formula
   Observation

   The slide shows $C = 180 - (52 + 20.5)$, then $B = 107.5$, and the final ratio uses sin 107.5° under c.

1. The displayed text labels the computed third angle as C in the equation line but as $B = 107.5$ in the next line; the intended meaning is the remaining angle, but the lettering is inconsistent on screen.

After obtaining one unknown angle in the SSA case, the video finds the last angle by subtracting the two known angles from 180°. It then returns to the Law of Sines ratio to solve for the remaining side if needed.

1. One unknown angle has already been found.

2. The triangle angle sum is 180°.

1. SSA step 1: solve for the angle opposite the known side
2. Finding the third angle from 180°

1. Audio
   Observation

   The speaker states that for an oblique triangle with angles A, B, C and opposite sides a, b, c, the relationship is a over sine A equals b over sine B equals c over sine C.

2. Formula
   Observation

   The displayed equation is $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.

In an oblique triangle with interior angles A, B, C and opposite side lengths a, b, c, the ratios $\frac{a}{\sin A}$, $\frac{b}{\sin B}$, and $\frac{c}{\sin C}$ are equal.

1. The figure is an oblique triangle.

2. A, B, C are the triangle’s interior angles.

3. a, b, c are the side lengths opposite A, B, C respectively.

For every oblique triangle satisfying the labeling condition, the three side-to-opposite-angle-sine ratios are equal.

1. Audio
   Observation

   The speaker says, "given two angles, we can always get the third. We just subtract these two angles from 180."

2. Formula
   Observation

   The screen computes $A = 180 - (85 + 50) = 45$.

If two interior angles of a triangle are known, the third interior angle is obtained by subtracting the sum of the two known angles from 180 degrees.

1. The figure is a triangle.

2. Two interior angle measures are known.

3. Angles are measured in degrees.

For any triangle, if two interior angles are known, the remaining interior angle is determined uniquely by the angle-sum relation.

1. Audio
   Observation

   The narrator says the triangle cannot exist if solving for the sine of another angle gives a value exceeding 1, 'which is impossible for the sine function.'

2. Formula
   Observation

   The slide text reads 'this will be the case if $\sin (A/B/C) > 1$'.

In an SSA setup, if the Law of Sines leads to $\sin (A) > 1$, $\sin (B) > 1$, or $\sin (C) > 1$ for the angle being solved, then no such triangle exists.

1. Two sides and one angle are given in an SSA configuration.

2. The Law of Sines is used to solve for another angle.

for the angle being solved in the SSA case

1. Audio
   Observation

   The narrator says, 'It is also possible that SSA triangles result in two solutions if we get a value for sine that could correspond to two different angles.'

2. Formula
   Observation

   The slide text reads 'we can sometimes get two solutions'.

An SSA triangle can have two solutions when the sine value obtained for an unknown angle corresponds to two different possible angles.

1. The problem is in side-side-angle form.

2. Solving with the Law of Sines yields a sine value compatible with more than one angle.

sometimes, for SSA triangles

1. Audio
   Observation

   The speaker introduces a side-angle-angle triangle and says the rest of the sides and angles can be obtained using the Law of Sines.

2. Formula
   Observation

   The screen substitutes the known values into $\frac{a}{\sin A} = \frac{7}{\sin 50^\circ} = \frac{c}{\sin 85^\circ}$.

3. Diagram
   Observation

   The triangle shows known angle 85°, known angle 50°, and known side 7 opposite 50°.

1. The clip stops before numerical evaluation of the final side lengths.

1. Expression

   $$\text{Known: } B = 50^\circ,\ C = 85^\circ,\ b = 7$$
   Explanation

   The example begins with two known angles and one known side, matching the SAA description spoken by the presenter.

   Justification

   Given data shown on the triangle diagram and stated in the audio.

   Shown in the video
2. Expression

   $$A = 180 - (85 + 50)$$
   Explanation

   The missing angle A is computed from the two known angles.

   Justification

   Triangle angle-sum relation as stated by the speaker.

   Shown in the video
3. Expression

   $$A = 45$$
   Explanation

   The arithmetic simplifies to the third angle measure.

   Justification

   Direct evaluation of 180 - 135.

   Shown in the video
4. Expression

   $$\frac{a}{\sin A} = \frac{7}{\sin 50^\circ} = \frac{c}{\sin 85^\circ}$$
   Explanation

   The Law of Sines is written with the known side 7 placed opposite the known angle 50°.

   Justification

   Substitution into the general formula $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.

   Shown in the video
5. Expression

   $$\frac{a}{\sin 45^\circ} = \frac{7}{\sin 50^\circ} = \frac{c}{\sin 85^\circ}$$
   Explanation

   The computed value $A = 45$ is substituted into the first denominator.

   Justification

   Replacement of A by its computed measure.

   Shown in the video
6. Expression

   $$a = 7\left(\frac{\sin 45^\circ}{\sin 50^\circ}\right)$$
   Explanation

   The equation is rearranged to isolate side a.

   Justification

   Algebraic multiplication of both sides by $\sin 45^\circ$.

   Shown in the video
7. Expression

   $$c = 7\left(\frac{\sin 85^\circ}{\sin 50^\circ}\right)$$
   Explanation

   The same rearrangement is applied to isolate side c.

   Justification

   Algebraic multiplication of both sides by $\sin 85^\circ$.

   Shown in the video

The clip derives explicit formulas for the two unknown sides in terms of the known side and the three angle sines, but does not display the final decimal values.

1. Formula
   Observation

   The screen shows the ratio $a/\sin 45$° = $7/\sin 50$° = $c/\sin 85$° and then the isolated formulas for a and c.

2. Animation
   Observation

   At about 5 seconds, the symbolic expressions are replaced by decimal answers $a = 6.46$ and $c = 9.10$, and the triangle labels update accordingly.

1. Expression

   $$\frac{a}{\sin 45^\circ}=\frac{7}{\sin 50^\circ}=\frac{c}{\sin 85^\circ}$$
   Explanation

   Set up the Law of Sines using the known side 7 opposite 50° and the two unknown sides opposite 45° and 85°.

   Justification

   Law of Sines pairs each side with the sine of its opposite angle.

   Shown in the video
2. Expression

   $$a=7\left(\frac{\sin 45^\circ}{\sin 50^\circ}\right)$$
   Explanation

   Isolate a by multiplying both sides of the first equality by sin 45°.

   Justification

   Algebraic rearrangement of the displayed proportion.

   Shown in the video
3. Expression

   $$c=7\left(\frac{\sin 85^\circ}{\sin 50^\circ}\right)$$
   Explanation

   Isolate c by multiplying both sides of the second equality by sin 85°.

   Justification

   Algebraic rearrangement of the displayed proportion.

   Shown in the video
4. Expression

   $$a=6.46,\quad c=9.10$$
   Explanation

   Evaluate the two expressions numerically to obtain approximate side lengths.

   Justification

   Calculator evaluation in degree mode, as stated by the narration.

   Shown in the video

The SAA example yields approximate side lengths $a = 6.46$ and $c = 9.10$.

1. Formula
   Observation

   The ASA slide shows $B = 180 - (95 + 35)$, then $B = 50$.

2. Audio
   Observation

   The narrator explains that in ASA we subtract the known angles from 180 to get the third.

1. Expression

   $$B=180-(95+35)$$
   Explanation

   Start from the two known angles 95° and 35° and subtract their sum from 180° to obtain the third angle.

   Justification

   Angle sum of a Euclidean triangle is 180°.

   Shown in the video
2. Expression

   $$B=50$$
   Explanation

   Compute the arithmetic result.

   Justification

   Direct subtraction.

   Shown in the video
3. Expression

   $$\frac{a}{\sin 35^\circ}=\frac{12}{\sin 50^\circ}=\frac{c}{\sin 95^\circ}$$
   Explanation

   Substitute $B = 50$ into the Law of Sines ratio so the known side 12 is paired with sin 50°.

   Justification

   Law of Sines after the third angle is known.

   Shown in the video

Once $B = 50$° is found, the ASA triangle is ready for direct Law of Sines substitution.

1. Formula
   Observation

   The SSA slide shows $A = \sin ^{-1}$(4 sin 52° / 9) = 20.5° and then $C = 180 - (52 + 20.5)$, $B = 107.5$.

2. Audio
   Observation

   The narrator says to first find the angle opposite the known side, then subtract the two angles from 180 to get the third.

1. The on-screen letters switch from C in the equation line to $B = 107.5$ in the next line, although the numerical meaning is the remaining angle.

1. Expression

   $$\frac{4}{\sin A}=\frac{9}{\sin 52^\circ}$$
   Explanation

   Use the known side-angle pair 9 opposite 52° and the known side 4 opposite the unknown angle A.

   Justification

   Law of Sines.

   Shown in the video
2. Expression

   $$A=\sin^{-1}\left(\frac{4\sin 52^\circ}{9}\right)=20.5^\circ$$
   Explanation

   Solve for A by taking the inverse sine of the rearranged ratio.

   Justification

   Algebraic isolation followed by inverse sine evaluation.

   Shown in the video
3. Expression

   $$180-(52+20.5)=107.5$$
   Explanation

   Subtract the two known angles from 180° to find the remaining angle.

   Justification

   Angle sum of a Euclidean triangle is 180°.

   Shown in the video
4. Expression

   $$\frac{4}{\sin 20.5^\circ}=\frac{9}{\sin 52^\circ}=\frac{c}{\sin 107.5^\circ}$$
   Explanation

   Update the Law of Sines ratio with all three angles now known so the remaining side c can be solved if needed.

   Justification

   Law of Sines after the third angle is determined.

   Shown in the video

The SSA example first produces $A = 20.5$°, then the remaining angle 107.5°, after which the last side can be found from the same ratio.

1. Audio
   Observation

   The narrator moves from 'one half base times height' to a formula using two sides and the sine of the angle when the height is unknown.

2. Formula
   Observation

   The slide changes from $A = 1/2(b)(h)$ to $\text{Area} = 1/2$ bc $\sin A = 1/2$ ab $\sin C = 1/2$ ac sin B.

1. The video does not show the intermediate derivation $h = b \sin A$ or an equivalent geometric proof inside this clip.

1. Expression

   $$A=\frac{1}{2}bh$$
   Explanation

   Begin with the standard area formula for any triangle using a base and its perpendicular height.

   Justification

   Known triangle area formula stated in the narration.

   Shown in the video
2. Expression

   $$\text{if } h \text{ is unknown, use two sides and the included angle}$$
   Explanation

   The narration motivates replacing the unavailable height with a trigonometric expression based on side-angle information.

   Justification

   Spoken explanation in the clip.

   Shown in the video
3. Expression

   $$\text{Area}=\frac{1}{2}bc\sin A=\frac{1}{2}ab\sin C=\frac{1}{2}ac\sin B$$
   Explanation

   Present the final sine-based area formulas for the three possible side pairs and their included angles.

   Justification

   Displayed formula on the final slide.

   Shown in the video

The clip introduces the sine-based oblique-triangle area formula as an alternative to base times height when the altitude is not known.

1. Audio
   Observation

   The speaker walks through a side-angle-angle triangle and says the remaining sides can be found using the Law of Sines.

2. Formula
   Observation

   The screen shows $A = 180 - (85 + 50)$, $A = 45$, then $a = 7(\sin 45^\circ / \sin 50^\circ)$ and $c = 7(\sin 85^\circ / \sin 50^\circ)$.

3. Diagram
   Observation

   The triangle is labeled with 85°, 50°, side 7, and unknown sides a and c.

1. No final numerical approximations for a and c appear before the clip ends.

Given a triangle with angles 85° and 50° and the side opposite the 50° angle equal to 7, find expressions for the remaining side lengths a and c.

1. One angle is 85°.

2. Another angle is 50°.

3. The side opposite the 50° angle has length 7.

4. The unknown sides are labeled a and c.

Use the Law of Sines to solve for the remaining sides after determining the third angle.

1. Expression

   $$A = 180 - (85 + 50)$$
   Explanation

   Find the missing angle from the two given angles.

   Justification

   Triangle angle sum property stated in the video.

   Shown in the video
2. Expression

   $$A = 45$$
   Explanation

   Evaluate the subtraction to obtain the third angle.

   Justification

   Arithmetic simplification.

   Shown in the video
3. Expression

   $$\frac{a}{\sin 45^\circ} = \frac{7}{\sin 50^\circ} = \frac{c}{\sin 85^\circ}$$
   Explanation

   Substitute the known angle-side pair and the known angle values into the Law of Sines.

   Justification

   Direct substitution into $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.

   Shown in the video
4. Expression

   $$a = 7\left(\frac{\sin 45^\circ}{\sin 50^\circ}\right)$$
   Explanation

   Isolate side a from the first equality.

   Justification

   Multiply both sides by $\sin 45^\circ$.

   Shown in the video
5. Expression

   $$c = 7\left(\frac{\sin 85^\circ}{\sin 50^\circ}\right)$$
   Explanation

   Isolate side c from the second equality.

   Justification

   Multiply both sides by $\sin 85^\circ$.

   Shown in the video

The unknown sides are expressed as $a = 7(\sin 45^\circ / \sin 50^\circ)$ and $c = 7(\sin 85^\circ / \sin 50^\circ)$.

The example is internally consistent with the displayed Law of Sines setup; however, the clip does not show a numerical check or final decimal answers.

1. Diagram
   Observation

   A triangle is shown with angles 45°, 50°, 85° and known side 7 opposite 50°.

2. Formula
   Observation

   The slide writes $a/\sin 45$° = $7/\sin 50$° = $c/\sin 85$° and then $a = 6.46$, $c = 9.10$.

Given a triangle with angles 45°, 50°, and 85°, and the side opposite 50° equal to 7, find the other two side lengths.

1. Angles: 45°, 50°, 85°

2. Known side: 7 opposite 50°

Compute the side lengths a and c.

1. Expression

   $$\frac{a}{\sin 45^\circ}=\frac{7}{\sin 50^\circ}=\frac{c}{\sin 85^\circ}$$
   Explanation

   Match each unknown side with its opposite angle and set all ratios equal to the known ratio.

   Justification

   Law of Sines.

   Shown in the video
2. Expression

   $$a=7\left(\frac{\sin 45^\circ}{\sin 50^\circ}\right)$$
   Explanation

   Solve the first equality for a.

   Justification

   Algebraic rearrangement.

   Shown in the video
3. Expression

   $$c=7\left(\frac{\sin 85^\circ}{\sin 50^\circ}\right)$$
   Explanation

   Solve the second equality for c.

   Justification

   Algebraic rearrangement.

   Shown in the video
4. Expression

   $$a=6.46,\quad c=9.10$$
   Explanation

   Evaluate numerically in degrees.

   Justification

   Calculator approximation as described by the narration.

   Shown in the video

$a = 6.46$ and $c = 9.10$ (approximate).

The displayed final slide replaces the symbolic expressions with these decimal values and labels the triangle accordingly.

1. Diagram
   Observation

   A triangle is shown with angles 95° and 35°, included side 12, and unknown angle B at the third vertex.

2. Formula
   Observation

   The slide computes $B = 180 - (95 + 35) = 50$ and then shows $a/\sin 35$° = $12/\sin 50$° = $c/\sin 95$°.

Given a triangle with two angles 95° and 35° and the included side 12, determine the third angle and set up the Law of Sines ratio for the remaining sides.

1. Angles: 95° and 35°

2. Included side: 12

Find the missing angle B and write the Law of Sines setup.

1. Expression

   $$B=180-(95+35)$$
   Explanation

   Subtract the known angles from 180° to obtain the third angle.

   Justification

   Triangle angle sum.

   Shown in the video
2. Expression

   $$B=50$$
   Explanation

   Compute the missing angle.

   Justification

   Arithmetic.

   Shown in the video
3. Expression

   $$\frac{a}{\sin 35^\circ}=\frac{12}{\sin 50^\circ}=\frac{c}{\sin 95^\circ}$$
   Explanation

   Substitute the known side 12 with its opposite angle 50° and pair the other sides with their opposite angles.

   Justification

   Law of Sines.

   Shown in the video

$B = 50$°, and the ratio is $a/\sin 35$° = $12/\sin 50$° = $c/\sin 95$°.

The slide text states that this ratio can be used to solve for the other two sides.

1. Diagram
   Observation

   A triangle is shown with sides 9 and 4 and angle 52° opposite side 9.

2. Formula
   Observation

   The slide computes $A = \sin ^{-1}$(4 sin 52° / 9) = 20.5°, then $180 - (52 + 20.5) = 107.5$, and updates the ratio to include sin 107.5°.

1. The on-screen labeling of the final computed angle is inconsistent between C and B, though the numerical value 107.5 is clear.

Given a triangle with sides 9 and 4 and the angle 52° opposite the side of length 9, find angle A opposite the side of length 4 and then the remaining angle.

1. Side 9 opposite 52°

2. Side 4 opposite unknown angle A

Compute A and the third angle, then set up the ratio for side c.

1. Expression

   $$\frac{4}{\sin A}=\frac{9}{\sin 52^\circ}$$
   Explanation

   Use the known side-angle pair to relate the unknown angle A to side 4.

   Justification

   Law of Sines.

   Shown in the video
2. Expression

   $$A=\sin^{-1}\left(\frac{4\sin 52^\circ}{9}\right)=20.5^\circ$$
   Explanation

   Solve for A using inverse sine.

   Justification

   Algebraic isolation and calculator evaluation.

   Shown in the video
3. Expression

   $$180-(52+20.5)=107.5$$
   Explanation

   Subtract the two known angles from 180° to get the remaining angle.

   Justification

   Triangle angle sum.

   Shown in the video
4. Expression

   $$\frac{4}{\sin 20.5^\circ}=\frac{9}{\sin 52^\circ}=\frac{c}{\sin 107.5^\circ}$$
   Explanation

   Now all three angles are known, so the last side c can be solved from the same ratio if needed.

   Justification

   Law of Sines.

   Shown in the video

$A = 20.5$°, remaining angle = 107.5°, and c is obtained from the updated Law of Sines ratio.

The slide explicitly displays these numerical substitutions in the final ratio.

1. Caption evidence
   Observation

   Problem statement: $a = 7$, $A = 32$°, $B = 61$°

2. Caption evidence
   Observation

   Solution revealed: $b = 11.6$, $c = 13.2$, $C = 87$°

For a triangle with sides $a/b/c$, and corresponding angles $A/B/C$, use the listed information to fill in the missing information: $a = 7$, $A = 32$°, $B = 61$°

1. $a = 7$

2. $A = 32$°

3. $B = 61$°

Find b, c, and C

1. Expression

   $$C = 180^\circ - 32^\circ - 61^\circ = 87^\circ$$
   Explanation

   Use the triangle angle sum theorem to find the third angle.

   Justification

   Sum of angles in a triangle is 180 degrees.

   Derived from the video
2. Expression

   $$\frac{7}{\sin 32^\circ} = \frac{b}{\sin 61^\circ} = \frac{c}{\sin 87^\circ}$$
   Explanation

   Set up the Law of Sines equation.

   Justification

   Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides.

   Derived from the video
3. Expression

   $$b = \frac{7 \sin 61^\circ}{\sin 32^\circ} \approx 11.6$$
   Explanation

   Solve for side b.

   Justification

   Algebraic manipulation of the Law of Sines equation.

   Derived from the video
4. Expression

   $$c = \frac{7 \sin 87^\circ}{\sin 32^\circ} \approx 13.2$$
   Explanation

   Solve for side c.

   Justification

   Algebraic manipulation of the Law of Sines equation.

   Derived from the video

$b = 11.6$, $c = 13.2$, $C = 87$°

Values match the solution revealed on screen.

1. Caption evidence
   Observation

   Problem statement: $a = 8.4$, $B = 55$°, $C = 79$°

2. Caption evidence
   Observation

   Solution revealed: $b = 9.6$, $c = 11.5$, $A = 46$°

For a triangle with sides $a/b/c$, and corresponding angles $A/B/C$, use the listed information to fill in the missing information: $a = 8.4$, $B = 55$°, $C = 79$°

1. $a = 8.4$

2. $B = 55$°

3. $C = 79$°

Find b, c, and A

1. Expression

   $$A = 180^\circ - 55^\circ - 79^\circ = 46^\circ$$
   Explanation

   Use the triangle angle sum theorem to find the first angle.

   Justification

   Sum of angles in a triangle is 180 degrees.

   Derived from the video
2. Expression

   $$\frac{8.4}{\sin 46^\circ} = \frac{b}{\sin 55^\circ} = \frac{c}{\sin 79^\circ}$$
   Explanation

   Set up the Law of Sines equation.

   Justification

   Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides.

   Derived from the video
3. Expression

   $$b = \frac{8.4 \sin 55^\circ}{\sin 46^\circ} \approx 9.6$$
   Explanation

   Solve for side b.

   Justification

   Algebraic manipulation of the Law of Sines equation.

   Derived from the video
4. Expression

   $$c = \frac{8.4 \sin 79^\circ}{\sin 46^\circ} \approx 11.5$$
   Explanation

   Solve for side c.

   Justification

   Algebraic manipulation of the Law of Sines equation.

   Derived from the video

$b = 9.6$, $c = 11.5$, $A = 46$°

Values match the solution revealed on screen.

1. Caption evidence
   Observation

   Problem statement: $a = 10.2$, $b = 4.8$, $A = 82$°

2. Caption evidence
   Observation

   Solution revealed: $c = 9.7$, $B = 27.8$°, $C = 70.2$°

For a triangle with sides $a/b/c$, and corresponding angles $A/B/C$, use the listed information to fill in the missing information: $a = 10.2$, $b = 4.8$, $A = 82$°

1. $a = 10.2$

2. $b = 4.8$

3. $A = 82$°

Find c, B, and C

1. Expression

   $$\frac{10.2}{\sin 82^\circ} = \frac{4.8}{\sin B}$$
   Explanation

   Set up the Law of Sines equation to solve for angle B.

   Justification

   Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides.

   Derived from the video
2. Expression

   $$\sin B = \frac{4.8 \sin 82^\circ}{10.2} \approx 0.466$$
   Explanation

   Solve for sin B.

   Justification

   Algebraic manipulation of the Law of Sines equation.

   Derived from the video
3. Expression

   $$B = \arcsin(0.466) \approx 27.8^\circ$$
   Explanation

   Solve for angle B.

   Justification

   Inverse sine function.

   Derived from the video
4. Expression

   $$C = 180^\circ - 82^\circ - 27.8^\circ = 70.2^\circ$$
   Explanation

   Use the triangle angle sum theorem to find the third angle.

   Justification

   Sum of angles in a triangle is 180 degrees.

   Derived from the video
5. Expression

   $$\frac{10.2}{\sin 82^\circ} = \frac{c}{\sin 70.2^\circ}$$
   Explanation

   Set up the Law of Sines equation to solve for side c.

   Justification

   Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides.

   Derived from the video
6. Expression

   $$c = \frac{10.2 \sin 70.2^\circ}{\sin 82^\circ} \approx 9.7$$
   Explanation

   Solve for side c.

   Justification

   Algebraic manipulation of the Law of Sines equation.

   Derived from the video

$c = 9.7$, $B = 27.8$°, $C = 70.2$°

Values match the solution revealed on screen.

1. Diagram
   Observation

   A right triangle labeled "Right Triangle" with a 90° marker appears first.

2. Diagram
   Observation

   It is replaced by a triangle labeled "Oblique Triangle" with the text "(no right angle)".

3. Audio
   Observation

   The speaker contrasts what was learned for right triangles with what can be done for other types of triangles.

1. Right triangle with 90° marker

2. Oblique triangle with vertices A, B, C and sides a, b, c

1. The visual focus shifts from a right triangle to an oblique triangle.

2. The label changes from "Right Triangle" to "Oblique Triangle (no right angle)".

1. Both visuals are triangular diagrams used to motivate trigonometric relationships.

The sequence establishes that the upcoming law is intended for triangles without a right angle, where the previously discussed right-triangle trigonometry no longer applies directly.

1. Animation
   Observation

   Purple circles highlight vertices A, B, C and then sides a, b, c.

2. Animation
   Observation

   Pink arrows point from each vertex toward the opposite side.

3. Audio
   Observation

   The speaker says the sides are precisely opposite their corresponding angles.

1. Triangle vertices A, B, C

2. Triangle sides a, b, c

3. Highlight circles

4. Directional arrows

1. Vertices are highlighted first, then sides.

2. Arrows indicate the correspondence from each angle to its opposite side.

1. The underlying triangle remains the same during the highlighting sequence.

The animation visually encodes the matching rule required by the Law of Sines: each side in a numerator must correspond to the sine of the angle opposite it.

1. Diagram
   Observation

   The worked triangle shows 85°, 50°, side 7, and unknown sides a and c.

2. Formula
   Observation

   The equation below updates from the general law to substituted values and then to isolated expressions for a and c.

3. Animation
   Observation

   A blue box highlights the known ratio $\frac{7}{\sin 50^\circ}$.

1. Triangle with labeled angles and side

2. Law of Sines equation

3. Highlighted known ratio box

4. Derived expressions for a and c

1. The generic formula is replaced by one containing the specific numbers 85°, 50°, and 7.

2. The missing angle A is computed and inserted as 45°.

3. The known ratio is emphasized, then the equations are rearranged to isolate a and c.

1. The same triangle and the same Law of Sines relationship are used throughout the example.

The visual progression demonstrates the solution method: identify a complete angle-side pair, substitute known values, and algebraically solve for each unknown side.

1. Animation
   Observation

   The SAA slide first shows symbolic expressions for a and c, then around 5 seconds replaces them with decimal values and updates the triangle labels.

1. Triangle with angles 45°, 50°, 85°

2. Side labels a, c, and known side 7

3. Law of Sines ratio line

4. Bottom solution line for a and c

1. The bottom line changes from $a = 7$(sin 45°/sin 50°), $c = 7$(sin 85°/sin 50°) to $a = 6.46$, $c = 9.10$.

2. The side labels on the triangle change from variables a and c to numeric values 6.46 and 9.10.

1. The angles 45°, 50°, and 85° remain fixed.

2. The known side 7 remains fixed.

3. The overall Law of Sines ratio structure remains the same.

The animation visually connects the algebraic Law of Sines setup to its numerical solution for the unknown sides.

1. Animation
   Observation

   A red circle highlights the unknown angle B, then the computed value 50° appears and the denominator sin B changes to sin 50°.

1. Triangle with angles 95°, 35°, and unknown B

2. Side label 12

3. Law of Sines ratio with sin B

1. The unknown angle B is highlighted.

2. The text $B = 180 - (95 + 35)$ and $B = 50$ appears.

3. The ratio updates from $12/\sin B$ to $12/\sin 50$°.

1. The known angles 95° and 35° stay fixed.

2. The known side 12 stays fixed.

The visual sequence shows that ASA problems first require recovering the third angle before the Law of Sines ratio can be fully instantiated.

1. Animation
   Observation

   The SSA slide first circles angle A and displays $A = 20.5$°, then circles the top vertex and displays the remaining angle 107.5°, and finally circles side c while the ratio updates to sin 107.5°.

1. The letter used for the remaining angle is inconsistent on screen between C and B, but the highlighted target and numerical value are clear.

1. Triangle with sides 9 and 4 and angle 52°

2. Unknown angle A

3. Remaining top angle

4. Unknown side c

5. Law of Sines ratio

1. Angle A is highlighted and assigned 20.5°.

2. The remaining angle is computed as 107.5° and inserted into the ratio.

3. Side c is highlighted as the next quantity to solve.

1. The given side lengths 9 and 4 remain fixed.

2. The given angle 52° remains fixed.

The animation encodes the SSA procedure order: first recover an opposite angle, then use angle sum, then return to the side ratio.

1. Animation
   Observation

   The slide changes from a triangle with base b and dashed height h and formula $A = 1/2(b)(h)$ to a generic triangle ABC with the sine-based area formula.

1. Oblique triangle with base b and height h

2. Generic triangle ABC

3. Area formulas

1. The height h is replaced by a question mark to indicate it may be unknown.

2. The formula changes from $A = 1/2(b)(h)$ to $\text{Area} = 1/2$ bc $\sin A = 1/2$ ab $\sin C = 1/2$ ac sin B.

1. The topic remains triangle area.

2. The triangle is still treated as an oblique triangle.

The visual shift motivates replacing an unavailable altitude with a formula built from two sides and their included angle.

1. Diagram
   Observation

   A static diagram of a triangle with vertices A, B, C and sides a, b, c is shown alongside the area formula.

1. Triangle ABC

2. Sides a, b, c

3. Angles A, B, C

4. Area formula text

1. The diagram remains static throughout the segment.

The diagram visually defines the variables used in the area formula and subsequent problems.

1. Diagram
   Observation

   A slide titled 'CHECKING COMPREHENSION' appears with three problems. After a pause, the solutions appear in red text below each problem.

1. Title text

2. Three problem statements

3. Three solution sets

1. Solutions appear in red text after a delay.

1. The problem statements remain visible.

This visual event presents practice problems for the viewer to solve using the Law of Sines, followed by the answers for self-checking.

1. Audio
   Observation

   The speaker says, "For these, all the SOHCAHTOA stuff we learned won't work anymore. But there are other relationships we can use, and one of them is called the Law of Sines."

2. Diagram
   Observation

   The oblique triangle is explicitly labeled as having no right angle.

One might assume the SOHCAHTOA relationships learned for right triangles can be used unchanged for any triangle.

The video states that for oblique triangles, the SOHCAHTOA approach no longer works, and introduces the Law of Sines as another relationship to use instead.

1. Audio
   Observation

   The narrator says that if the sine value exceeds 1, the triangle cannot exist because that is impossible for the sine function.

2. Formula
   Observation

   The slide states 'this will be the case if $\sin (A/B/C) > 1$'.

One might continue treating an SSA computation as valid even when the solved sine value is greater than 1.

The video states that $\sin (A/B/C) > 1$ is impossible for the sine function, so the corresponding triangle is invalid and cannot exist.

1. Audio
   Observation

   The narrator says SSA triangles can result in two solutions if a sine value corresponds to two different angles.

2. Formula
   Observation

   The slide text reads 'we can sometimes get two solutions'.

One might assume that two sides and a non-included angle determine a unique triangle.

The video explicitly notes that SSA can produce two solutions when the obtained sine value matches two different possible angles.

1. Audio
   Observation

   The narrator contrasts the usual base-times-height formula with the case 'if we don't know the height'.

2. Formula
   Observation

   The slide replaces $A = 1/2(b)(h)$ with $\text{Area} = 1/2$ bc $\sin A = 1/2$ ab $\sin C = 1/2$ ac sin B.

One might think the area of an oblique triangle cannot be found unless the perpendicular height is known.

The video introduces an alternative formula using two sides and the sine of the included angle, so the height need not be given directly.

1. Audio
   Observation

   The speaker defines oblique triangles and then immediately introduces the Law of Sines as a relationship for them.

2. Formula
   Observation

   The law is displayed under the heading "Understanding The Law of Sines" with an oblique triangle diagram.

The Law of Sines is presented as the tool for handling oblique triangles, i.e., triangles without a right angle.

1. Diagram
   Observation

   A right triangle is shown first, then an oblique triangle labeled "(no right angle)".

2. Audio
   Observation

   The speaker contrasts prior learning about right triangles with the need to handle other triangle types.

The video contrasts oblique triangles with right triangles to explain why a new relationship is needed beyond SOHCAHTOA.

1. Caption evidence
   Observation

   The section title reads "Using The Law of Sines for SAA Triangles".

2. Audio
   Observation

   The speaker says the remaining sides and angles can be obtained using the Law of Sines.

The SAA configuration is used as the concrete setting in which the Law of Sines is applied.

1. Formula
   Observation

   The screen first computes $A = 45$, then substitutes that value into the Law of Sines equation.

2. Audio
   Observation

   The speaker says the third angle is found first, then the angle-side pair is used to get the other sides.

In the worked example, computing the missing angle is a prerequisite step that creates the full set of angle measures needed to apply the Law of Sines effectively.

1. Formula
   Observation

   The SAA example is solved directly by writing $a/\sin 45$° = $7/\sin 50$° = $c/\sin 85$°.

The SAA method is a direct application of the Law of Sines ratio to recover unknown side lengths.

1. Audio
   Observation

   The narrator says that in ASA we subtract the known angles from 180 to get the third, and once we have that we are ready to plug values into the law of sines.

2. Formula
   Observation

   The slide computes $B = 50$ before replacing sin B with sin 50° in the ratio.