Warm-Up Exercises
ANSWER 8.0
ANSWER 11.3
2. If m P = 58° and r = 5, find p.
1. If PR = 12 and m R = 19°, find p.
Use this diagram for Exercises 1–4.
Warm-Up Exercises
4. If r = 8 and p = 12, find q.
ANSWER 14.4
ANSWER 10.4
3. If m P = 60°, and p = 9 , find q.
Use this diagram for Exercises 1–4.
Warm-Up ExercisesEXAMPLE 1 Use an inverse tangent to find an angle measure
Use a calculator to approximate the measure of A to the nearest tenth of a degree.
SOLUTION
Because tan A =
1520
34
= = 0.75, tan–1 0.75 = m A
Use a calculator.
tan –1 0.75 36.86989765. . .
ANSWERSo, the measure of A is approximately 36.9o.
Warm-Up ExercisesEXAMPLE 2 Use an inverse sine and an inverse cosine
Let A and B be acute angles in a right triangle. Use a calculator to approximate the measures of A and B to the nearest tenth of a degree.
a. sin A = 0.87 b. cos B = 0.15
SOLUTION
a. m A = sin –1 0.87 60.5o
b. m B = cos –1 0.15 81.4o
Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2
1. Look back at Example 1. Use a calculator and an inverse tangent to approximate m C to the nearest tenth of a degree.
ANSWER 53.1o
2. Find m D to the nearest tenth of a degree if sin D = 0.54.
ANSWER 32.7o
Warm-Up ExercisesEXAMPLE 3 Solve a right triangle
Solve the right triangle. Round decimal answers to the nearest tenth.
SOLUTION
STEP 1Find m B by using the Triangle Sum Theorem.180o = 90o + 42o + m B
48o = m B
Warm-Up ExercisesEXAMPLE 3 Solve a right triangle
STEP 2Approximate BC by using a tangent ratio.
tan 42o = BC70
Write ratio for tangent of 42o.
70 tan 42o = BC Multiply each side by 70.
Approximate tan 42o70 0.9004 BC
63 BC Simplify and round answer.
Warm-Up ExercisesEXAMPLE 3 Solve a right triangle
STEP 3Approximate 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.
Divide each side by cos 42o.
Use a calculator to find cos 42o.
AB 70 cos 42o=
AB 70 0.7431
AB 94.2 Simplify .
ANSWER The angle measures are 42o, 48o, and 90o. Theside lengths are 70 feet, about 63 feet, and about 94 feet.
Warm-Up ExercisesEXAMPLE 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?
Warm-Up ExercisesEXAMPLE 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.
Warm-Up ExercisesGUIDED PRACTICE for Examples 3 and 4
3. Solve a right triangle that has a 40o angle and a 20 inch hypotenuse.
ANSWER
40o, 50o, and 90o, about 12.9 in., about 15.3 in. and 20 in.
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.
No; the rake is 5.7° so it is slightly larger than the suggested range.
ANSWER
Warm-Up ExercisesDaily Homework Quiz
ANSWER 54.9°
Use this diagram for exercises 1-3.
1. If x = 9 and z = 11 find m X to the nearest tenth of a degree.
ANSWER 63.4°
2. If y = 5 and z = 12 find m X to the nearest tenth of a degree.
Warm-Up ExercisesDaily Homework Quiz
ANSWER
The angles are 17.4°, 72.6°, and 90°; the sides are 12, about 3.6, and about 11.5.
3. If m Y = 17.4° and z = 12 solve ∆XYZ.
Use this diagram for exercises 1-3.
