Question 1

Which of the following is the correct formula for the derivative of $$f(x) = \sin(x) \cos(x)$$?

Accepted Answer

$$\cos^2$(x) - \sin^2$(x)$$

Question 2

Given $$\sin(\alpha) = \dfrac{3}{5}$$ and $$\alpha is in $$the first quadrant, what is $$\cos(\alpha)$$?

Accepted Answer

$$\dfrac{4}{5}$$

Question 3

Solve for $$x in $$the interval $$[0, 2\pi)$$: $$2\sin(x) - 1 = 0.$$

Accepted Answer

Question 4

Find all solutions to $$	an(x) = 1$$ for $$x in$$ $$[0, 2\pi).$$

Accepted Answer

$$x = \dfrac{\pi}{4}, \dfrac{5\pi}{4}$$

Question 5

If $$\sin(\alpha) = \dfrac{3}{5}$$ and $$\cos(\beta) = \dfrac{5}{13}$$, with $$\alpha, \beta in $$the first quadrant, what is $$\sin(\alpha + \beta)$$?

Accepted Answer

$$\dfrac{56}{65}$$

Question 6

Which of the following is an identity?

Accepted Answer

$$\sin^2$(x) + \cos^2$(x) = 1$$

Question 7

Find the critical numbers of $$f(x) = \sin(x) on $$the interval $$(0, 2\pi).$$

Accepted Answer

$$x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}$$

Question 8

Given $$f(x) = 	an(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\sec^2$(x)$$

Question 9

If $$\sin(	heta) = -\dfrac{1}{2}$$ and $$	heta is in$$ $$[0, 2\pi)$$, what are all possible values of $$	heta$$?

Accepted Answer

$$\dfrac{7\pi}{6}, \dfrac{11\pi}{6}$$

Question 10

Which of the following is the correct formula for $$\cos(\alpha - \beta)$$?

Accepted Answer

$$\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)$$

Question 11

A Ferris wheel rotates such that the height $$h (in $$meters) of a passenger above the ground at time $$t (in $$seconds) is given by $$h(t) = 10 + 8\sin\left(\dfrac{\pi}{15}t\right). $$What is the maximum height reached?

Accepted Answer

$$18$$ meters

Question 12

If $$f(x) = \sin(x) + \cos(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\cos(x) - \sin(x)$$

Question 13

Which of the following is the general solution to $$\sin(x) = 0$$?

Accepted Answer

$$x = n\pi$$, where $$n is $$any integer

Question 14

Given $$\cos(	heta) = -\dfrac{3}{5}$$ and $$	heta is in $$the second quadrant, what is $$\sin(	heta)$$?

Accepted Answer

$$\dfrac{4}{5}$$

Question 15

Which of the following is the correct derivative of $$f(x) = \sin^2$(x)$$?

Accepted Answer

$$2\sin(x)\cos(x)$$

Question 16

If $$	an(x) = \dfrac{1}{\sqrt{3}}$$, what is the general solution for $$x$$?

Accepted Answer

$$x = \dfrac{\pi}{6} + n\pi$$, where $$n is $$any integer

Question 17

A sound wave can be modeled by $$y(t) = 0.5\sin(440\pi t). $$What is the period of this wave?

Accepted Answer

$$\dfrac{1}{220}$$ seconds

Question 18

Which of the following is the correct formula for $$\sin(\alpha - \beta)$$?

Accepted Answer

$$\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)$$

Question 19

If $$f(x) = \cos(x)$$, what is the value of $$f'(\pi)$$?

Accepted Answer

$$0$$

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