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Example 1. SOLUTION. Use the law of cosines to find the length b . You need to find b , so use. â€“. b 2. a 2. +. c 2. 2 ac cos B. =. (. ). 7. Write law of cosines. (. ). â€“. b 2. a 2. +. c 2. 2 a c cos B. =. â€“. Substitute for a , c , and B. b 2. 7 2. +. 12 2.

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## Example 1

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**Example 1**SOLUTION Use the law of cosines to find the length b. You need to find b, so use . – b2 a2 + c2 2ac cos B = ( ) 7 Write law of cosines. ( ) – b2 a2 + c2 2accosB = – Substitute for a, c, and B. b2 72 + 122 cos41° 2 12 = – Evaluate powers and multiply. b2 49 + 144 168 cos41° = Solve for a Side (SAS) Find the length b given that a7, c12, and B41°. = = =**Example 1**b 8.1 ≈ ≈ 66.2 Take positive square root. b2 66.2 ≈ Solve for a Side (SAS) Simplify.**Checkpoint**3.6 ANSWER 1. 10.1 ANSWER 2. Solving for a Side Find the unknown side length of the triangle to the nearest tenth.**Checkpoint**8.2 ANSWER 3. Solving for a Side Find the unknown side length of the triangle to the nearest tenth.**Example 2**– Write law of cosines. b2 a2 + c2 2accos B = – a2 c2 b2 + Rewrite formula, by solving for cosB. cos B = 2ac ( ( ) ) 5 8 – 52 82 112 + Substitute for a, b, and c. cos B = 2 – Simplify using a calculator. cos B 0.4 = Solve for a Angle (SSS) Find the measure of angle B. Use the law of cosines to find the measure of angle B.**Example 2**( ) – B cos 1 0.4 113.6° ≈ = – Solve for a Angle (SSS) Use the inverse cosine function to find the angle measure.**Checkpoint**4. Find the measure of angle C given that a3, b7, and c9. Sketch the triangle. = = = about 123.2° ANSWER 5. Find the measure of angle A given that a4, b8, and c6. Sketch the triangle. = = = about 29° ANSWER Solving for an Angle**Example 3**Find the length a given that b5, B28°, and C110°. = = = – – A 180° 110° 28° 42° = = a b = sin A sin B Write law of sines. a 5 Substitute for A, B, and b. = sin42° sin28° Choose a Method 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.**Example 3**5 sin 42° a = sin 28° Simplify using a calculator. a 7.1 ≈ Choose a Method Solve for the variable.**Example 4**a. What is the distance from A to B? b. Find the measures of the angles in the triangle. 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.**Example 4**SOLUTION a. You know the lengths of two sides and their included angle. So, use the law of cosines. Write law of cosines. – c2 a2 + b2 2ab cos C = Substitute for a, b and C. ( ) ( ) – c2 1252 + 1402 cos44° 2 125 140 = Evaluate powers. – c2 15,625 + 19,600 35,000 cos44° = Simplify. c2 10,048.1 ≈ Take positive square root. c 100.2 ≈ ≈ 10,048.1 Use the Law of Sines and the Law of Cosines**Example 4**ANSWER The distance from A to B is about 100 feet. b. To find the measures of the angles in the triangle, use the law of sines. sin A sin C = Write law of sines. a c sin A sin 44° Substitute for a, c, and C. = 125 100.2 125 sin 44° Multiply each side by 125. sin A = 100.2 Use the Law of Sines and the Law of Cosines**Example 4**( ) A sin 1 0.8666 60.1° ≈ = – You then know that . – – B 180° 44° 60.1° 75.9° ≈ = ANSWER The measure of angle A is about 60° and the measure of angle B is about 76°. Simplify using a calculator. sin A 0.8666 ≈ Use the Law of Sines and the Law of Cosines Use the inverse sine function to find the angle measure.**Checkpoint**6. ANSWER a7.8, B30°, b5.1 = = = 7. ANSWER H 42.5°,G 112.5,g 10.9 ≈ ≈ ≈ Use the Law of Sines and the Law of Cosines Use any method to find the unknown angle measures and side lengths.**Checkpoint**8. r 6.6,S ANSWER 63.4°, T 36.6° = = = Use the Law of Sines and the Law of Cosines Use any method to find the unknown angle measures and side lengths.

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