Q: A tower stands on level ground. From a point 80 feet from the base, the angle of elevation to the top of the tower is 85.4°. What is the height $$h of $$the tower? (Refer to the first diagram.)

$$h = 80 \tan 85.4^\circ \approx 994 ft$$

Q: You are standing 800 feet from Mt. Rushmore. The angle of elevation to the bottom of Lincoln’s face is 32°, and to the top is 35°. What is the height of the sculpture? (Refer to the third diagram.)

$$800 \tan 35^\circ - 800 \tan 32^\circ \approx 60.3 ft$$

Q: A hot-air balloon rises vertically. From a point 125 feet away, the angle of elevation changes from 19.2° to 31.7°. How far does the balloon rise during this period?

$$125 \tan 31.7^\circ - 125 \tan 19.2^\circ \approx 33.7 ft$$

Question 1

A tower stands on level ground. From a point 80 feet from the base, the angle of elevation to the top of the tower is 85.4°. What is the height $$h of $$the tower? (Refer to the first diagram.)

Accepted Answer

$$h = 80 	an 85.4^\circ \approx 994 ft$$

Question 2

A guy wire is attached from the ground to a pole at a point 6.7 yards above the ground. The wire is 13.8 yards long. What is the angle $$A$$ the wire makes with the ground? (Refer to the second diagram.)

Accepted Answer

$$A = \sin$$^{-1}$$ \left( \frac{6.7}{13.8} \right) \approx 29.0^\circ$$

Question 3

You are standing 800 feet from Mt. Rushmore. The angle of elevation to the bottom of Lincoln’s face is 32°, and to the top is 35°. What is the height of the sculpture? (Refer to the third diagram.)

Accepted Answer

$$800 	an 35^\circ - 800 	an 32^\circ \approx 60.3 ft$$

Question 4

A wheelchair ramp is 23 feet long and rises to a height of 6 feet. What is the angle of elevation of the ramp, to the nearest tenth of a degree?

Accepted Answer

$$\sin$$^{-1}$$ \left( \frac{6}{23} \right) \approx 15.1^\circ$$

Question 5

A hot-air balloon rises vertically. From a point 125 feet away, the angle of elevation changes from 19.2° to 31.7°. How far does the balloon rise during this period?

Accepted Answer

$$125 	an 31.7^\circ - 125 	an 19.2^\circ \approx 33.7 ft$$

Question 6

A jet flies 5 miles on a bearing of N 35° E, then turns 90° and flies 7 miles on a bearing of S 55° E. What is the bearing of the jet from its starting point?

Accepted Answer

N 89.5° E

Question 7

A ball on a string is pulled 8 inches below its rest position and released. The period for the motion is 2 seconds. What is the equation for the ball’s simple harmonic motion?

Accepted Answer

$$d = -8 \cos \pi t$$

Question 8

An object moves in simple harmonic motion described by $$d = 5 \cos \frac{\pi}{2} t. $$What is the maximum displacement, frequency, and period?

Accepted Answer

Maximum displacement: 5 inches; Frequency: $$\frac{1}{4}$$ oscillation per second; Period: 4 seconds

Question 9

Evaluate $$\cos(\sin$$^{-1}$$ \frac{4}{5}).$$

Accepted Answer

$$\frac{3}{5}$$

Question 10

Evaluate $$\csc[\cos$$^{-1}$$(-\frac{\sqrt{3}}{2})].$$

Accepted Answer

$$-\frac{2}{\sqrt{3}}$$

Question 11

If $$x \u003e 0$$, write $$	an(\cos$$^{-1}$$ x) as an $$algebraic expression in $$x.$$

Accepted Answer

$$\frac{\sqrt{1-x^2$}}{x}$$

Question 12

A ball on a string is pulled 6 inches below its rest position and released. The period for the motion is 4 seconds. What is the equation for the ball’s simple harmonic motion?

Accepted Answer

$$d = -6 \cos \frac{\pi}{2} t$$

Question 13

An object moves in simple harmonic motion described by $$d = 12 \cos \frac{\pi}{4} t. $$What is the frequency and period?

Accepted Answer

Frequency: $$\frac{1}{8}$$ oscillation per second; Period: 8 seconds

Question 14

A ship sails 2.3 miles on a bearing of S 31° W, then turns 90° clockwise and sails 3.5 miles on a bearing of N 59° W. What is the bearing from the starting point?

Accepted Answer

S 87.7° W

Question 15

Evaluate $$	an \left( \frac{\pi}{6} ight).$$

Accepted Answer

$$\frac{1}{\sqrt{3}}$$

Question 16

Evaluate $$\sin \left( \frac{3\pi}{4} \right).$$

Accepted Answer

$$\frac{\sqrt{2}}{2}$$

Question 17

Evaluate $$\cos \left( -\frac{\pi}{6} \right).$$

Accepted Answer

$$\frac{\sqrt{3}}{2}$$

Question 18

Evaluate $$	an \left( \frac{3\pi}{4} ight).$$

Accepted Answer

$$-1$$

Applications of Trigonometric Functions: Solving Right Triangles, Bearings, and Simple Harmonic Motion

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