1. OBJECTIVE:TO SOLVE RIGHT TRIANGLE PROBLEMS USING THE TRIG FUNCTIONS. USING THE RATIOS UNIT 10: SECTION 8.7

2. EXAMPLE 4 Find a hypotenuse using an angle of depression SKIING You are skiing on a mountain with an altitude of 1200 meters. The angle of depression is 21 o. About how far do you ski down the mountain?

3. EXAMPLE 4 Find a hypotenuse using an angle of depression SOLUTION sin 21 o Write ratio for sine of 21 o. sin 21 o Substitute. x sin 21 o = 1200 Multiply each side by x. x = 1200. sin 21 o Divide each side by sin 21 o x 1200. 0.3584 Use a calculator to find sin 21 o x 3348.2 Simplify. opp hyp = 1200 x = ANSWER You ski about 3348 meters down the mountain.

4. GUIDED PRACTICE for Example 4 6 Suppose the angle of depression in Example 4 is 28°. About how far would you ski? WHAT IF ? SOLUTION sin 28 o Write ratio for sine of 28 o. sin 28 o Substitute. x sin 28 o = 1200 Multiply each side by x. x = 1200. sin 28 o Divide each side by sin 28 o opp hyp = 1200 x =

5. GUIDED PRACTICE for Examples 4 x 1200. 0.4694 Use a calculator to find sin 28 o x 2556m Simplify.

6. EXAMPLE 3 Use a trigonometric ratio to find a hypotenuse DOG RUN You want to string cable to make a dog run from two corners of a building, as shown in the diagram. Write and solve a proportion using a trigonometric ratio to approximate the length of cable you will need.

7. EXAMPLE 3 Use a trigonometric ratio to find a hypotenuse SOLUTION sin 35 o = opp hyp Write ratio for sine of 35 o. sin 35 o = 11 x Substitute. x sin 35 o = 11 Multiply each side by x. x = 11. sin 35 o Divide each side by tan. 35 o x 11. 0.5736 Use a calculator to find tan. 35 o x 19.2 Simplify. ANSWER You will need a little more than 19 feet of cable.

8. EXAMPLE 3 Solve a right triangle Solve the right triangle. Round decimal answers to the nearest tenth. SOLUTION STEP 1 Find m B by using the Triangle Sum Theorem. 180 o = 90 o + 42 o + m B 48 o = m B

9. EXAMPLE 3 Solve a right triangle STEP 2 Approximate BC by using a tangent ratio. tan 42 o = BC 70 Write ratio for tangent of 42 o. 70 tan 42 o = BC Multiply each side by 70. Approximate tan. 42 o 70 0.9004 BC 63 BC Simplify and round answer.

10. EXAMPLE 3 Solve a right triangle STEP 3 Approximate AB by using a cosine ratio. cos 42 o = 70 AB Write ratio for cosine of 42 o. AB cos 42 o = 70 Multiply each side by AB. Divide each side by cos. 42 o Use a calculator to find cos. 42 o AB 70 cos 42 o = AB 70 0.7431 AB 94.2 Simplify. ANSWER The angle measures are 42 o, 48 o, and 90 o. The side lengths are 70 feet, about 63 feet, and about 94 feet.

11. EXAMPLE 4 Solve a real-world problem THEATER DESIGN Suppose your school is building a raked stage. The stage will be 30 feet long from front to back, with a total rise of 2 feet. A rake (angle of elevation) of 5 o or less is generally preferred for the safety and comfort of the actors. Is the raked stage you are building within the range suggested?

12. EXAMPLE 4 Solve a real-world problem SOLUTION Use the sine and inverse sine ratios to find the degree measure x of the rake. sin x o = opp. hyp = 2 30 0.0667 x sin –1 0.0667 3.842 ANSWER The rake is about 3.8 o, so it is within the suggested range of 5 o or less.