EXAMPLE 3 Solve a right triangle

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Solving Right Triangles Essential Question How do I solve a right triangle?

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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. 180o = 90o + 42o + m B 48o = m B

Approximate BC by using a tangent ratio. EXAMPLE 3 Solve a right triangle STEP 2 Approximate BC by using a tangent ratio. tan 42o = BC70 Write ratio for tangent of 42o. 70 tan 42o = BC Multiply each side by 70. 70 0.9004 BC Approximate tan. 42o 63 BC Simplify and round answer.

Approximate AB by using a cosine ratio. EXAMPLE 3 Solve a right triangle STEP 3 Approximate AB by using a cosine ratio. cos 42o = 70 AB Write ratio for cosine of 42o. AB cos 42o = 70 Multiply each side by AB. AB 70 cos 42o = Divide each side by cos. 42o AB 70 0.7431 Use a calculator to find cos. 42o AB 94.2 Simplify . ANSWER The angle measures are 42o, 48o, and 90o. 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 5o 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. hyp 2 30 0.0667 x sin –1 0.0667 3.842 ANSWER The rake is about 3.8o, so it is within the suggested range of 5o or less.

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

Approximate YZ by using a sine ratio. GUIDED PRACTICE for Examples 3, and 4 STEP 2 Approximate YZ by using a sine ratio. sin 40o = XY20 Write ratio for sine of 40o. 20 sin 40o = XY Multiply each side by 20. 20 0.6428 XY Approximate sin. 40o 12.9 BC Simplify and round answer.

Approximate AB by using a cosine ratio. GUIDED PRACTICE for Examples 3, and 4 STEP 3 Approximate AB by using a cosine ratio. cos 40o = YZ 20 Write ratio for cosine of 40o. 20 cos 40o = YZ Multiply each side by 20. 20 0.7660 YZ Approximate cos. 40o 15.3 YZ Simplify and round answer. ANSWER The angle measures are 40o, 50o, and 90o. 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. hyp 2 20 0.1 x sin –1 0.1 5.739 No; the rake is 5.7° so it is slightly larger than the suggested range. ANSWER