Warm-Up Exercises ABC Find the unknown parts of . A = 75°, B 82°, c 16 1. ANSWER C = 23°, a 39.6, ≈ b 40.6 2. B = 131°, b 52, c 38 ANSWER C 33.5°, a 18.4 ≈ A 15.5°,

Warm-Up Exercises 3. In a triangle, if and can you find the measure of angle C? Explain. b = 7, c 15, B 75°, = ANSWER No; using the law of sines you get so no such triangle exists. sin C 2.07

Find the length b given that a 7, c 12, and B 41°. = = = Example 1 Solve for a Side (SAS) Find the length b given that a 7, c 12, and B 41°. = = = SOLUTION Use the law of cosines to find the length b. You need to find b, so use . b 2 = a 2 + c 2 – 2ac cos B b 2 = a 2 + c 2 – 2ac cos B Write law of cosines. 2 ( ) 7 12 b 2 = 72 + 122 – cos 41° Substitute for a, c, and B. b 2 = 49 + 144 – 168 cos 41° Evaluate powers and multiply.

Example 1 b 2 66.2 ≈ b ≈ 66.2 8.1 Solve for a Side (SAS) Simplify. Take positive square root. b ≈ 66.2 8.1 4

Find the unknown side length of the triangle to the nearest tenth. Checkpoint Solving for a Side Find the unknown side length of the triangle to the nearest tenth. 1. ANSWER 3.6 2. ANSWER 10.1

Find the unknown side length of the triangle to the nearest tenth. Checkpoint Solving for a Side Find the unknown side length of the triangle to the nearest tenth. 3. ANSWER 8.2

Find the measure of angle B. Example 2 Solve for a Angle (SSS) Find the measure of angle B. Use the law of cosines to find the measure of angle B. b 2 = a 2 + c 2 – 2ac cos B Write law of cosines. Rewrite formula, by solving for cos B. 2ac a 2 + c 2 b 2 – = cos B Substitute for a, b, and c. 52 + 82 112 – = cos B 2 ( ) 5 8 = cos B 0.4 – Simplify using a calculator.

Use the inverse cosine function to find the angle measure. Example 2 Solve for a Angle (SSS) Use the inverse cosine function to find the angle measure. ( ) 0.4 cos 1 – = 113.6° ≈ B 8

Checkpoint Solving for an Angle 4. Find the measure of angle C given that a 3, b 7, and c 9. Sketch the triangle. = ANSWER about 123.2° = 5. Find the measure of angle A given that a 4, b 8, and c 6. Sketch the triangle. ANSWER about 29°

Find the length a given that b 5, B 28°, and C 110°. = Example 3 Choose a Method Find the length a given that b 5, B 28°, and C 110°. = You know two angles and one side. Use the law of sines. Use the fact that the sum of the angle measures is 180° to find A. A 180° 110° 42° = – 28° Write law of sines. = sin A a sin B b Substitute for A, B, and b. = sin 42° a sin 28° 5 10

Example 3 = sin 28° 5 sin 42° a a 7.1 ≈ Choose a Method Solve for the variable. a 7.1 ≈ Simplify using a calculator. 11

Surveyor A bridge is being built Example 4 Use the Law of Sines and the Law of Cosines Surveyor A bridge is being built across a river. A surveyor needs to measure the distance from point A to point B. The surveyor is at point C and measures an angle of 44°. The surveyor measures the distances from point C to points A and B and finds the distances to be 140 feet and 125 feet. a. What is the distance from A to B? b. Find the measures of the angles in the triangle. 12

Example 4 Use the Law of Sines and the Law of Cosines You know the lengths of two sides and their included angle. So, use the law of cosines. SOLUTION a. c 2 = a 2 + b 2 – 2ab cos C Write law of cosines. 2 ( ) 125 140 c 2 = 1252 + 1402 – cos 44° Substitute for a, b and C. Evaluate powers. c 2 = 15,625 + 19,600 – cos 44° 35,000 c 2 10,048.1 ≈ Simplify. Take positive square root. c ≈ 10,048.1 100.2 13

The distance from A to B is about 100 feet. Example 4 Use the Law of Sines and the Law of Cosines ANSWER The distance from A to B is about 100 feet. To find the measures of the angles in the triangle, use the law of sines. b. = c sin C a sin A Write law of sines. = 100.2 sin 44° 125 sin A Substitute for a, c, and C. = 100.2 125 sin 44° sin A Multiply each side by 125. 14

Use the inverse sine function to find the angle measure. Example 4 Use the Law of Sines and the Law of Cosines sin A 0.8666 ≈ Simplify using a calculator. Use the inverse sine function to find the angle measure. ( ) 0.8666 sin 1 – = 60.1° ≈ A You then know that . B 180° 44° 75.9° – = 60.1° ≈ ANSWER The measure of angle A is about 60° and the measure of angle B is about 76°. 15

Use any method to find the unknown angle measures and side lengths. Checkpoint Use the Law of Sines and the Law of Cosines Use any method to find the unknown angle measures and side lengths. 6. ANSWER a 7.8, B 30°, b 5.1 = 7. ANSWER ≈ H 42.5°, G 112.5, g 10.9

Use any method to find the unknown angle measures and side lengths. Checkpoint Use the Law of Sines and the Law of Cosines Use any method to find the unknown angle measures and side lengths. 8. ANSWER r 6.6, S = 63.4°, 36.6° T