Question 1

A pendulum swings through an angle of $$40^\circ$$ each second. If the pendulum is 45 inches long, how far does its tip move each second? (Round your answer to two decimal places.)

Accepted Answer

$$32.71 in.$$

Question 2

Given $$	an 	heta = \dfrac{12}{5}$$ and $$\dfrac{\pi}{2} \u003c 	heta \u003c \dfrac{3\pi}{2}$$, find $$\cos 	heta.$$

Accepted Answer

$$-\dfrac{5}{13}$$

Question 3

Find the exact value of $$\csc(960^\circ).$$

Accepted Answer

$$-2$$

Question 4

A twenty-five foot ladder just reaches the top of a house and forms an angle of $$41.5^\circ$$ with the wall of the house. How tall is the house? (Round your answer to the nearest 0.1 foot.)

Accepted Answer

$$18.7 ft$$

Question 5

Given $$\cos 	heta = \dfrac{1}{4}$$ and $$\csc 	heta \u003e 0$$, find $$\sin \dfrac{	heta}{2}.$$

Accepted Answer

$$\dfrac{8 - 2\sqrt{15}}{4}$$

Question 6

Find the exact value of $$\sin\left(2\cos$$^{-1}$$\left(-\dfrac{3}{5}\right)\right).$$

Accepted Answer

$$\dfrac{24}{25}$$

Question 7

Given $$\csc 	heta = -\dfrac{7}{4}$$ with $$	heta in $$quadrant III, find $$\cot 	heta.$$

Accepted Answer

$$-\dfrac{33}{7}$$

Question 8

Solve the equation on the interval $$0 \leq 	heta \u003c 2\pi$$: $$\sin(2	heta) + \sin 	heta = 0.$$

Accepted Answer

$$\left\{0, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}\right\}$$

Question 9

Write the trigonometric expression $$\cos(\sin$$^{-1}$$ u) as an $$algebraic expression in $$u.$$

Accepted Answer

$$\sqrt{1-u^2$}$$

Question 10

Find the inverse function $$f^{-1} of$$ $$f(x) = 5\sin x - 3.$$

Accepted Answer

$$f^{-1}(x$$) = \$$sin^{-1}$$\left(\dfrac{x+3}{5}\right)$$

Question 11

Given $$\sin \alpha = -\dfrac{3}{5}$$ with $$\dfrac{3\pi}{2} \u003c \alpha \u003c 2\pi$$ and $$	an \beta = -\dfrac{12}{5}$$ with $$\dfrac{\pi}{2} \u003c \beta \u003c \pi$$, find $$\cos(\alpha + \beta).$$

Accepted Answer

$$\dfrac{16}{65}$$

Question 12

Find the phase shift of the function $$y = 3\cos\left(-3x - \dfrac{\pi}{2}\right).$$

Accepted Answer

$$\dfrac{\pi}{6}$$ units$$\;to $$the right

Question 13

Establish the identity: $$(1-\cos x)(1+\cos x) = \sin^2$ x$$

Accepted Answer

True, because $$1-\cos^2$ x = \sin^2$ x$$

Question 14

Solve the equation on $$0 \leq 	heta \u003c 2\pi$$: $$(\csc 	heta - 2)(\cot 	heta + 1) = 0.$$

Accepted Answer

$$\left\{\dfrac{\pi}{6}, \dfrac{3\pi}{4}, \dfrac{5\pi}{6}, \dfrac{7\pi}{4}\right\}$$

Question 15

A surveyor selects a point $$C$$ that is $$55$$ feet from point $$A$$ and $$72$$ feet from point $$B. $$The angle $$ACB is$$ $$53^\circ. $$How far apart are points $$A$$ and $$B$$? (Round to the nearest tenth of a foot.)

Accepted Answer

$$113.9 ft$$

Question 16

Given $$\cos 	heta = \dfrac{15}{17}$$ and $$\dfrac{3\pi}{2} \u003c 	heta \u003c 2\pi$$, find $$\cot 	heta.$$

Accepted Answer

$$-\dfrac{8}{15}$$

Question 17

Find the exact value of $$\sin\left(-\dfrac{11\pi}{12}\right).$$

Accepted Answer

$$-\dfrac{\sqrt{6} + \sqrt{2}}{4}$$

Question 18

Find the exact value of $$\cos\left(	an$$^{-1}$$\dfrac{4}{3} - \sin$$^{-1}$$\dfrac{3}{5}ight).$$

Accepted Answer

$$\dfrac{24}{25}$$

Question 19

Given $$\csc 	heta = \dfrac{5}{3}$$ and $$\dfrac{\pi}{3} \u003c 	heta \u003c \pi$$, find $$\cos(2	heta).$$

Accepted Answer

$$\dfrac{7}{25}$$

Precalculus Exam 3 Study Guide – Step-by-Step Guidance

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