Q: From a point 200 feet in front of a library, the angle of elevation to the top of the flagpole is $$30^\circ$$ and to the base of the flagpole is $$20^\circ. $$What is the height of the flagpole?

$$200 \left( \tan 30^\circ - \tan 20^\circ \right)$$

Q: Solve the equation $$\sin^2$\theta = \cos^2$\theta = 1 + \cos\theta$$ for $$\theta in$$ $$[0, 2\pi).$$

$$\theta = 0, \frac{2\pi}{3}, \frac{4\pi}{3}$$

Question 1

A radio tower is 500 feet tall and a guy wire is attached to the top and anchored to the ground at a distance of 80 feet from the base. What is the length of the guy wire?

Accepted Answer

$$\sqrt{500^2$ + 80^2$}$$

Question 2

Given $$a = 3$$, $$b = 7$$, and $$A = 12^\circ$$, how many triangles can be formed and what is the measure of angle $$B$$?

Accepted Answer

Two triangles; $$B \approx 49^\circ$$

Question 3

From a point 200 feet in front of a library, the angle of elevation to the top of the flagpole is $$30^\circ$$ and to the base of the flagpole is $$20^\circ. $$What is the height of the flagpole?

Accepted Answer

$$200 \left( 	an 30^\circ - 	an 20^\circ ight)$$

Question 4

What is the exact value of $$\cos\left(\frac{5\pi}{6}\right)$$?

Accepted Answer

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

Question 5

Write the equation of a sine function with amplitude 2, period $$\pi$$, and phase shift $$\frac{\pi}{2}.$$

Accepted Answer

$$2\sin\left(2x - 1\right)$$

Question 6

Find the exact value of $$\cos\left(22.5^\circ\right)$$ using a half-angle formula.

Accepted Answer

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

Question 7

Given $$f(x) = \frac{3x}{x-6}$$ and $$g(x) = \frac{6}{x}$$, what is the domain of $$f(g(x))$$?

Accepted Answer

$$x 
eq 0, x 
eq 6$$

Question 8

Solve the equation $$2^x + 2$$^{x-2}$$ = 9.0$$ for $$x.$$

Accepted Answer

$$x = 3$$

Question 9

Find the inverse function of $$f(x) = \frac{2x+3}{x-2}.$$

Accepted Answer

$$f^{-1}(x$$) = \frac{2x+3}{x-2}$$

Question 10

Given $$f(x) = \frac{6}{x^2$-3x-6}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 3, x = -2$$

Question 11

If $$f(x) = -x^2$(x+3)(x-9)$$, what is the y-intercept?

Accepted Answer

$$y = 0$$

Question 12

Given $$y = 2\cos(\frac{\pi}{x} - 2) + 5$$, what is the amplitude?

Accepted Answer

$$2$$

Question 13

What is the domain of $$g(x) = \ln\left(\frac{x}{x-3}\right)$$?

Accepted Answer

$$(-\infty, 0) \cup (3, \infty)$$

Question 14

Solve the equation $$\sin^2$	heta = \cos^2$	heta = 1 + \cos	heta$$ for $$	heta in$$ $$[0, 2\pi).$$

Accepted Answer

$$	heta = 0, \frac{2\pi}{3}, \frac{4\pi}{3}$$

Question 15

Given $$f(x) = -3x^2$ - 6x + 9$$, what is the axis of symmetry?

Accepted Answer

$$x = -1$$

Question 16

Given $$f(x) = \frac{6}{x^2$-3x-6}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 17

Given $$f(x) = x^2$(x+2)(x-4)$$, what is the degree of the polynomial?

Accepted Answer

$$4$$

Question 18

Given $$f(x) = x^2$(x+2)(x-4)$$, what are the zeros?

Accepted Answer

$$x = 0, x = -2, x = 4$$

Question 19

Given $$f(x) = \frac{6}{x^2$-3x-6}$$, what is the y-intercept?

Accepted Answer

$$y = -1$$

Precalculus Study Guide: Trigonometry, Functions, and Logarithms

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