1. Geometry tan A 20 21 === opposite adjacent BC AC tan B 21 20 === opposite adjacent AC BC Write the tangent ratios for A and B. Lesson 8-3 The Tangent Ratio Additional Examples

2. Geometry To measure the height of a tree, Alma walked 125 ft from the tree and measured a 32° angle from the ground to the top of the tree. Estimate the height of the tree. The tree forms a right angle with the ground, so you can use the tangent ratio to estimate the height of the tree. tan 32° = height 125 Use the tangent ratio. height = 125 (tan 32°)Solve for height. 125 32 78.108669 Use a calculator. The tree is about 78 ft tall. Lesson 8-3 The Tangent Ratio Additional Examples

3. Geometry Find m R to the nearest degree. tan R = 47 41 Find the tangent ratio. So m R 49. Lesson 8-3 The Tangent Ratio m R tan –1 Use the inverse of the tangent. 47 41 Use a calculator. 48.900494 47 41 Additional Examples

4. Geometry Use the triangle to find sin T, cos T, sin G, and cos G. Write your answer in simplest terms. Lesson 8-4 Sine and Cosine Ratios sin T = = 12 20 3535 = opposite hypotenuse cos T = = 16 20 4545 = adjacent hypotenuse sin G = = 16 20 4545 = opposite hypotenuse cos G = = 12 20 3535 = adjacent hypotenuse Additional Examples

5. Geometry 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. Lesson 8-4 Sine and Cosine Ratios 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. Additional Examples

6. Geometry 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 Lesson 8-4 Sine and Cosine Ratios Additional Examples

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

8. Geometry Describe 1 and 2 as they relate to the situation shown. One side of the angle of depression is a horizontal line. 1 is the angle of depression from the airplane to the building. One side of the angle of elevation is a horizontal line. 2 is the angle of elevation from the building to the airplane. Lesson 8-5 Angles of Elevation and Depression Additional Examples

9. Geometry A surveyor stands 200 ft from a building to measure its height with a 5-ft tall theodolite. The angle of elevation to the top of the building is 35°. How tall is the building? Draw a diagram to represent the situation. Use a calculator.200 35 140.041508 So x 140. Lesson 8-5 Angles of Elevation and Depression To find the height of the building, add the height of the Theodolite, which is 5 ft tall. The building is about 140 ft + 5 ft, or 145 ft tall. x = 200 tan 35° Solve for x. Use the tangent ratio.tan 35° = x 200 Additional Examples

10. Geometry An airplane flying 3500 ft above ground begins a 2° descent to land at an airport. How many miles from the airport is the airplane when it starts its descent? Draw a diagram to represent the situation. Use the sine ratio.sin 2° = 3500 x x = 3500 sin 2° Solve for x. 3500 2 100287.9792 Use a calculator. 5280 18.993935 Divide by 5280 to convert feet to miles. The airplane is about 19 mi from the airport when it starts its descent. Lesson 8-5 Angles of Elevation and Depression Additional Examples