EXAMPLE 3 Solve a right triangle Solve the

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 48 o = m B B

EXAMPLE 3 Solve a right triangle STEP 2 Approximate BC by using a tangent ratio. tan 42 o = BC 70 70 tan 42 o = BC Write ratio for tangent of 42 o. Multiply each side by 70. 70 0. 9004 BC Approximate tan. 42 o 63 BC Simplify and round answer.

EXAMPLE 3 Solve a right triangle STEP 3 Approximate AB by using a cosine ratio. cos 42 o = 70 Write ratio for cosine of 42 o. AB AB cos 42 o = 70 70 AB = cos 42 o 70 AB 0. 7431 AB 94. 2 Multiply each side by AB. Divide each side by cos. 42 o Use a calculator to find cos. 42 o 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.

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?

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 xo = opp. = 2 hyp 30 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.

GUIDED PRACTICE for Examples 3, and 4 3. Solve a right triangle that has a 40 o angle and a 20 inch hypotenuse. SOLUTION X STEP 1 Find m X by using the Triangle Sum Theorem. 180 o = 90 o + 40 o + m 50 o = m X X Y Z

GUIDED PRACTICE for Examples 3, and 4 STEP 2 Approximate YZ by using a sine ratio. sin 40 o = XY 20 20 sin 40 o = XY Write ratio for sine of 40 o. Multiply each side by 20. 20 0. 6428 XY Approximate sin. 40 o 12. 9 BC Simplify and round answer.

GUIDED PRACTICE for Examples 3, and 4 STEP 3 Approximate AB by using a cosine ratio. cos 40 o = YZ 20 Write ratio for cosine of 40 o. 20 cos 40 o = YZ Multiply each side by 20. 20 0. 7660 YZ Approximate cos. 40 o 15. 3 YZ Simplify and round answer. ANSWER The angle measures are 40 o, 50 o, and 90 o. The side lengths are 12. 9 in. , about 15. 3 in. , and 20 in.

GUIDED PRACTICE for Examples 3, and 4 4. WHAT IF? In Example 4, suppose another raked stage is 20 feet long from front to back with a total rise of 2 feet. Is this raked stage safe? Explain. sin xo = opp. = 2 hyp 20 x sin – 1 0. 1 5. 739 ANSWER No; the rake is 5. 7° so it is slightly larger than the suggested range.