Q: A building 160 feet tall casts a 60-foot-long shadow. If a person stands at the end of the shadow and looks up to the top of the building, with their eyes 5 feet above ground, what is the angle $$\theta of $$elevation to the top of the building? (Leave in calculator-ready form.)

$$\theta = \tan$$^{-1}$$(\frac{155}{60})$$

Q: A radio tower is built in two sections. From a point 87 ft from the base, the angle of elevation to the top of the first section is $$28^\circ$$ and to the top of the second section is $$45^\circ. $$What is the height $$h of $$the top section? (Leave in calculator-ready form.)

$$h = 87(\tan 45^\circ - \tan 28^\circ)$$

Q: From the top of a roller coaster, 60 yards above the ground, a rider looks down and sees the merry-go-round and the Ferris wheel. If the angles of depression are $$11^\circ$$ and $$8^\circ$$ respectively, how far is the rider from the merry-go-round? (Leave in calculator-ready form.)

$$d = 60(\tan 82^\circ - \tan 79^\circ)$$

Question 1

What is the range of the secant function $$y = \sec x$$?

Accepted Answer

$$(−\infty, −1] \cup [1, ∞)$$

Question 2

Find the exact value of $$\sin$$^{-1}$$(\sin \frac{3\pi}{2}).$$

Accepted Answer

$$-\frac{\pi}{2}$$

Question 3

Given a right triangle with $$a = 3$$ and $$c = 7$$, find angle $$A.$$

Accepted Answer

$$\sin$$^{-1}$$(\frac{3}{7})$$

Question 4

What is the domain of $$y = \cos$$^{-1}$$ x$$?

Accepted Answer

$$[−1,1]$$

Question 5

Find the phase shift of $$y = -4 \cos(-2x + \frac{\pi}{8}).$$

Accepted Answer

$$\frac{\pi}{16}$$

Question 6

What is the range of $$y = 	an$$^{-1}$$ x$$?

Accepted Answer

$$(−\frac{\pi}{2}, \frac{\pi}{2})$$

Question 7

Given $$P = (\frac{3}{5}, -\frac{4}{5}) is a $$point on the unit circle corresponding to real number $$t$$, find $$\cos t$$ and $$\cot t.$$

Accepted Answer

$$\cos t = \frac{3}{5}, \cot t = -\frac{3}{4}$$

Question 8

A building 160 feet tall casts a 60-foot-long shadow. If a person stands at the end of the shadow and looks up to the top of the building, with their eyes 5 feet above ground, what is the angle $$	heta of $$elevation to the top of the building? (Leave in calculator-ready form.)

Accepted Answer

$$	heta = 	an$$^{-1}$$(\frac{155}{60})$$

Question 9

A radio tower is built in two sections. From a point 87 ft from the base, the angle of elevation to the top of the first section is $$28^\circ$$ and to the top of the second section is $$45^\circ. $$What is the height $$h of $$the top section? (Leave in calculator-ready form.)

Accepted Answer

$$h = 87(	an 45^\circ - 	an 28^\circ)$$

Question 10

A forest ranger sights a fire directly to the south. A second ranger, 6 miles directly east, sights the fire at a bearing of $$S32^\circ W. $$How far is the first ranger from the fire? (Leave in calculator-ready form.)

Accepted Answer

$$x = 6 	an 58^\circ$$

Question 11

A jet leaves a runway on a bearing of $$N25^\circ E$$ from the control tower. After flying 5 miles, it turns $$90^\circ$$ and flies on a bearing of $$S55^\circ E$$ for 5 miles. What is the bearing from the control tower to the jet? (Leave in calculator-ready form.)

Accepted Answer

$$N70^\circ E$$

Question 12

A news helicopter is 1500 ft directly above the finish line of a marathon. Two runners to the right of the helicopter sight the helicopter with angles of elevation $$35^\circ$$ and $$67^\circ$$, respectively. How far apart are the runners? (Leave in calculator-ready form.)

Accepted Answer

$$d = 1500(\cot 35^\circ - \cot 67^\circ)$$

Question 13

From the top of a roller coaster, 60 yards above the ground, a rider looks down and sees the merry-go-round and the Ferris wheel. If the angles of depression are $$11^\circ$$ and $$8^\circ$$ respectively, how far is the rider from the merry-go-round? (Leave in calculator-ready form.)

Accepted Answer

$$d = 60(	an 82^\circ - 	an 79^\circ)$$

Question 14

Two boats leave a dock together, one at 22 mph on a bearing of $$N31^\circ W$$ and the other at 16 mph on a bearing of $$S59^\circ W. $$How far apart are they after 30 minutes? (Round to the nearest tenth of a mile.)

Accepted Answer

$$d = \sqrt{185}$$ miles

Question 15

Explain why we restrict the domain of $$y = \cos x to$$ $$[0, \pi]$$ when defining its inverse.

Accepted Answer

Because $$y = \cos x is $$not one-to-one on $$\mathbb{R}$$, so we restrict to $$[0, \pi]$$ where it passes the horizontal line test and covers the full range $$[−1,1].$$

Question 16

True or False: $$\sin$$^{-1}$$(−\frac{1}{2})$$ and $$\cos$$^{-1}$$(−\frac{1}{2})$$ are both negative.

Accepted Answer

False, only $$\sin$$^{-1}$$(−\frac{1}{2}) is $$negative; $$\cos$$^{-1}$$(−\frac{1}{2}) is $$positive.

Question 17

What is the amplitude, period, and phase shift of $$y = 4 \cos(\frac{1}{2}x + \frac{\pi}{4})$$?

Accepted Answer

Amplitude: 4, Period: $$4\pi$$, Phase shift: $$-\frac{\pi}{2}$$

Question 18

Given $$y = \sin$$^{-1}$$ x$$, what are the input and output, and how is $$y = \sin x$$ used in creating them?

Accepted Answer

Input: ratios in $$[−1,1]$$, Output: angles in $$[−\frac{\pi}{2}, \frac{\pi}{2}]$$; $$y = \sin x is $$used by switching input and output and restricting the domain.

Precalculus Exam 2 Review: Step-by-Step Guidance with Visuals

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