Sine and Cosine Ratios May 9, 2003 Lesson 8-4 Check Skills You’ll Need (For help, go to Lesson 8-3.) For each triangle, find (a) the length of the leg opposite B and (b) the length of the leg adjacent to B. 1. 3. 2. Check Skills You’ll Need 8-4

b. The leg adjacent to B is the one that is a side of the angle: 12. Sine and Cosine Ratios May 9, 2003 Lesson 8-4 Check Skills You’ll Need 1. a. The leg opposite B is the one that is not a side of the angle: 9. b. The leg adjacent to B is the one that is a side of the angle: 12. 2. a. The leg opposite B is the one that is not a side of the angle: 7. b. The leg adjacent to B is the one that is a side of the angle: 2 78. 3. a. The leg opposite B is the one that is not a side of the angle: 10. b. The leg adjacent to B is the one that is a side of the angle: 3 29. Solutions 8-4

Use the figure for Exercises 1–3. The Tangent Ratio May 9, 2003 Lesson 8-3 Lesson Quiz Use the figure for Exercises 1–3. 1. Write the tangent ratio for K. 2. Write the tangent ratio for M. 3. Find m M to the nearest degree. 15 8 8 15 28 Find x to the nearest whole number. 4. 5. 52 29 8-4

Textbook

Textbook

Sine and Cosine Ratios May 9, 2003 Lesson 8-4 Notes One way to describe the relationship of sine and cosine is to say that sin x = cos(90 - x) for values of x between 0 and 90. This type of equation is called an identity because it is true for all the allowed values of the variable. 8-4

Writing Sine and Cosine Ratios May 9, 2003 Lesson 8-4 Additional Examples Writing Sine and Cosine Ratios Use the triangle to find sin T, cos T, sin G, and cos G. Write your answer in simplest terms. sin G = = 16 20 4 5 opposite hypotenuse cos T = = 16 20 4 5 adjacent hypotenuse cos G = = 12 20 3 5 adjacent hypotenuse sin T = = 12 20 3 5 opposite hypotenuse Quick Check 8-4

Real-World Connection Sine and Cosine Ratios May 9, 2003 Lesson 8-4 Additional Examples Real-World Connection A 20-ft. wire supporting a flagpole forms a 35˚ angle with the flagpole. To the nearest foot, how high is the flagpole? The flagpole, wire, and ground form a right triangle with the wire as the hypotenuse. Because you know an angle and the measures of its adjacent side and the hypotenuse, you can use the cosine ratio to find the height of the flagpole. cos 35° = height 20 Use the cosine ratio. height = 20 • cos 35° Solve for height. 20 35 16.383041 Use a calculator. The flagpole is about 16 ft tall. Quick Check 8-4

Using the Inverse of Sine and Cosine Sine and Cosine Ratios May 9, 2003 Lesson 8-4 Additional Examples Using the Inverse of Sine and Cosine A right triangle has a leg 1.5 units long and hypotenuse 4.0 units long. Find the measures of its acute angles to the nearest degree. Draw a diagram using the information given. Use the inverse of the cosine function to find m A. cos A = 1.5 4.0 0.375 = Use the cosine ratio. Use the inverse of the cosine. m A = cos–1(0.375) Use a calculator. 0.375 67.975687 Round to the nearest degree. m A 68 8-4

Definition of complementary angles Sine and Cosine Ratios May 9, 2003 Lesson 8-4 Additional Examples (continued) To find m B, use the fact that the acute angles of a right triangle are complementary. m A + m B = 90 Definition of complementary angles Substitute. 68 + m B 90 m B 22 The acute angles, rounded to the nearest degree, measure 68 and 22. Quick Check 8-4

Use this figure for Exercises 1 and 2. Sine and Cosine Ratios May 9, 2003 Lesson 8-4 Lesson Quiz Use this figure for Exercises 1 and 2. 1. Write the ratios for sin A and sin B. 2. Write the ratios for cos A and cos B. sin A = or , sin B = or 16 34 30 8 17 15 cos A = or , cos B = or 16 34 30 15 17 8 Use this figure for Exercises 3 and 4. 3. Find x to the nearest tenth. 4. Find y to the nearest tenth. 21.0 13.6 Use this figure for Exercises 5 and 6. 5. Find x to the nearest degree. 6. Find y to the nearest degree. 44 46 8-4