Q: What is the domain of the function $$y = \tan x$$?

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

Q: A ship sails 10 km east, then turns $$30^\circ$$ north of east and sails another 15 km. What is the ship's total displacement from its starting point?

$$\sqrt{10^2$ + 15^2$ - 2 \times 10 \times 15 \times \cos 30^\circ}$$

Question 1

Which of the following is the correct equation for a sine function with amplitude 3, period $$\pi$$, and a vertical shift up 2 units?

Accepted Answer

Question 2

A cosine function is given by $$y = 2 \cos\left(\frac{\pi}{3}x\right) - 1. $$What is the period of this function?

Accepted Answer

$$6$$

Question 3

Which transformation is applied to the graph of $$y = \sin x to $$obtain $$y = \sin(x - \frac{\pi}{4}) + 3$$?

Accepted Answer

Shift right $$\frac{\pi}{4}$$ units and up 3 units

Question 4

What is the domain of the function $$y = 	an x$$?

Accepted Answer

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

Question 5

Which of the following best describes the graph of $$y = \csc x$$?

Accepted Answer

It has vertical asymptotes at $$x = n\pi$$, where $$n is $$any integer

Question 6

A Ferris wheel rotates with a radius of 20 meters and completes one revolution every 60 seconds. If the lowest point is 2 meters above the ground, which equation models the height $$h (in $$meters) of a passenger above the ground as a function of time $$t (in $$seconds)?

Accepted Answer

$$h(t) = 20 \sin\left(\frac{2\pi}{60}t\right) + 22$$

Question 7

What is the exact value of $$\arccos\left(\frac{1}{2}\right)$$?

Accepted Answer

$$\frac{\pi}{3}$$

Question 8

If $$	heta = \arcsin\left(\frac{3}{5}ight)$$, what is $$\cos	heta$$?

Accepted Answer

$$\frac{4}{5}$$

Question 9

Which of the following is the range of $$y = \arctan x$$?

Accepted Answer

$$(-\frac{\pi}{2}, \frac{\pi}{2})$$

Question 10

A ship sails 10 km east, then turns $$30^\circ$$ north of east and sails another 15 km. What is the ship's total displacement from its starting point?

Accepted Answer

$$\sqrt{10^2$ + 15^2$ - 2 	imes 10 	imes 15 	imes \cos 30^\circ}$$

Question 11

Which of the following is the correct equation for a tangent function with period $$\pi$$ and vertical shift down 4 units?

Accepted Answer

$$y = 	an(x) - 4$$

Question 12

If $$y = \cos x is $$shifted up 5 units and reflected across the x-axis, what is the new equation?

Accepted Answer

$$y = -\cos x + 5$$

Question 13

What is the domain of $$y = \arcsin x$$?

Accepted Answer

$$[-1, 1]$$

Question 14

A mass on a spring oscillates according to $$y = 4 \sin\left(\frac{\pi}{2} t\right). $$What is the period of the motion?

Accepted Answer

$$4$$

Question 15

If $$y = \sec x$$, at which values of $$x$$ does the function have vertical asymptotes?

Accepted Answer

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

Question 16

Find the exact value of $$\arctan(1).$$

Accepted Answer

$$\frac{\pi}{4}$$

Question 17

A right triangle has legs of length 7 and 24. What is the measure of the angle opposite the leg of length 7?

Accepted Answer

$$\arctan\left(\frac{7}{24}\right)$$

Question 18

Which of the following is the range of $$y = \arccos x$$?

Accepted Answer

$$[0, \pi]$$

Question 19

A pilot flies 100 miles north, then 100 miles east. What is the bearing from the starting point to the final location?

Accepted Answer

$$45^\circ$$ north of east

Question 20

If $$y = \cot x$$, what is the period of the function?

Accepted Answer

$$\pi$$

Graphs and Applications of Trigonometric Functions

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