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Trigonometry : Law of Cosines and Law of Sines

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Example Question #1 : Law Of Sines

Find the length of the line segment $\overline{AB}$ in the triangle below.

Round to the nearest hundredth of a centimeter.

Triangle

Possible Answers:

$12.21\ cm$

$14.66\ cm$

$15.60\ cm$

$10.83\ cm$

$13.84\ cm$

Correct answer:

$13.84\ cm$

Explanation:

The law of sines states that

$\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}$ .

In this triangle, we are looking for the side length c, and we are given angle A, angle B, and side b. The sum of the interior angles of a triangle is $180^{\circ}$ ; using subtraction we find that angle C = $57^{\circ}$ .

We can now form a proportion that includes only one unknown, c:

$\frac{sin52^{\circ}}{13}=\frac{sin57^{\circ}}{c}$

Solving for c, we find that

$c=\frac{13sin57^{\circ}}{sin52^{\circ}}\approx13.84$ .

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Example Question #1 : Law Of Sines

In the triangle below, $m\angle A=78^\circ$ , $m\angle B=40^\circ$ , and c=14 . What is the length of side to the nearest tenth?

Triangle abc

Possible Answers:

19.2

15.5

9.2

12.6

10.2

Correct answer:

10.2

Explanation:

First, find $m\angle C$ . The sum of the interior angles of a triangle is $180^\circ$ , so $m\angle C = 180^\circ - 78^\circ - 40^\circ$ , or $62^\circ$ .

Using this information, you can set up a proportion to find side b:

$\frac{sinC}{c}=\frac{sinB}{b}$

$\frac{sin62^\circ}{14}=\frac{sin40^\circ}{b}$

$b=\frac{14sin40^\circ}{sin62^\circ}$

$b\approx10.2$

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Example Question #1 : Law Of Sines

In the triangle below, $m\angle A=75^\circ$ , $m\angle B=21^\circ$ , and c=19.3 .

Triangle abc

What is the length of side a to the nearest tenth?

Possible Answers:

7.2

7.0

18.7

19.9

52.0

Correct answer:

18.7

Explanation:

To use the law of sines, first you must find the measure of $\angle C$ . Since the sum of the interior angles of a triangle is $180^\circ$ , $m\angle C=84^\circ$ .

Law of sines:

$\frac{\sin A}{a}=\frac{\sin C}{c}$

$\frac{\sin 75^\circ}{a}=\frac{\sin 84^\circ}{19.3}$

$a=\frac{19.3 \sin 75^\circ}{\sin 84^\circ}$

$a\approx 18.7$

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Example Question #11 : Law Of Sines

Construction02

The triangle above has side lengths 3, 4, and 6. The angle opposite the side of length 6 measures 117.28 degrees, rounded to the nearest hundredth. Angle $\theta$ is opposite the side of length 3. What is the measure of $\theta$ , rounded to the nearest hundredth of a degree?

Possible Answers:

$7.63^\circ$

$26.38^\circ$

$44.51^\circ$

$15.03^\circ$

Correct answer:

$26.38^\circ$

Explanation:

Because the angle with measure 117.28 degrees is opposite the side of length 6, and angle $\theta$ is opposite the side of length 3, we can use the Law of Sines to solve for the measure of $\theta$ .

$\frac{\sin(117.28^{\circ})}{6}=\frac{\sin(\theta)}{3}$

$\theta=\arcsin(\frac{3\sin(117.28^\circ)}{6})$

$\theta=26.38^\circ$

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Example Question #11 : Law Of Sines

Find the length of side a using the law of sines. All angles are in degrees.

Tri

Possible Answers:

10.91

9.17

32.8

9.46

10.57

Correct answer:

9.46

Explanation:

The law of sines states that, given a triangle with sides a, b, and c and angles A, B, and C opposite to the corresponding sides,

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Tri

Since the sides and angles given are directly opposite, we can use the law of sines.

$\frac{a}{\sin 55^{\circ}}=\frac{10}{\sin 60^{\circ}}$

Solving for a, we get

$a=\frac{10\sin 55^{\circ}}{\sin 60}$

Evaluating the expression, we find that a=9.46

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Example Question #41 : Law Of Cosines And Law Of Sines

Find the measure of angle A.

Tri3

Possible Answers:

$49^{\circ}$

$66^{\circ}$

$60^{\circ}$

$47.5^{\circ}$

$87^{\circ}$

Correct answer:

$66^{\circ}$

Explanation:

The law of sines states that, given a triangle, the following relationship is always true:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

where a, b, and c are sides and A, B, and C are the angles opposite to the sides.

This problem does not give us the length of the side opposite to the angle we want to find, so we have to find it indirectly.

We start by finding the measure of the unmarked angle, which we'll represent as B:

$\frac{10}{\sin B}=\frac{12}{\sin 65^{\circ}}$

Solving for $\sin B$ , we get

$\frac{10\sin 65^{\circ}}{12}=\sin B$

$B=\arcsin(0.755)=49^\circ$

Now that we have the measures of two angles, we can find the measure of the third by the $180^{\circ}$ theorem:

$65^\circ+49^\circ+A=180^\circ$

$A=180^{\circ}-65^\circ-49^\circ=66^\circ$

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Example Question #11 : Law Of Sines

Find the length of side .

Possible Answers:

6.61

6.33

5.80

7.19

Correct answer:

6.33

Explanation:

Recall the Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Start by finding the value of angle :

A+56+39=180

A=85

Plug in the given values into the Law of Sines:

$\frac{4}{\sin 39}=\frac{a}{\sin 85}$

Rearrange the equation to solve for :

$a=(\sin 85)\frac{4}{\sin 39}=6.33$

Make sure to round to two places after the decimal.

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Example Question #81 : Triangles

Find the length of side

Possible Answers:

1.09

3.37

2.87

2.60

Correct answer:

2.87

Explanation:

Recall the Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Start by finding the value of angle :

C+36+76=180

C=68

Plug in the given values into the Law of Sines:

$\frac{3}{\sin 76}=\frac{c}{\sin 68}$

Rearrange the equation to solve for :

$c=(\sin 68)\frac{3}{\sin 76}=2.87$

Make sure to round to two places after the decimal.

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Example Question #41 : Law Of Cosines And Law Of Sines

Find the length of side .

Possible Answers:

8.44

9.06

8.07

10.27

Correct answer:

8.44

Explanation:

Recall the Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Start by finding the value of angle :

C+65+42=180

C=73

Plug in the given values into the Law of Sines:

$\frac{8}{\sin 65}=\frac{c}{\sin 73}$

Rearrange the equation to solve for :

$c=(\sin 73)\frac{8}{\sin 65}=8.44$

Make sure to round to two places after the decimal.

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Example Question #11 : Law Of Sines

Find the length of .

Possible Answers:

11.49

12.51

11.28

12.22

Correct answer:

11.49

Explanation:

Recall the Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Start by finding the value of angle :

C+71+28=180

C=81

Plug in the given values into the Law of Sines:

$\frac{11}{\sin 71}=\frac{c}{\sin 81}$

Rearrange the equation to solve for :

$c=(\sin 81)\frac{11}{\sin 71}=11.49$

Make sure to round to two places after the decimal.

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Trigonometry : Law of Cosines and Law of Sines

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Law Of Sines

Example Question #1 : Law Of Sines

Example Question #1 : Law Of Sines

Example Question #11 : Law Of Sines

Example Question #11 : Law Of Sines

Example Question #41 : Law Of Cosines And Law Of Sines

Example Question #11 : Law Of Sines

Example Question #81 : Triangles

Example Question #41 : Law Of Cosines And Law Of Sines

Example Question #11 : Law Of Sines

All Trigonometry Resources

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