Q: Which of the following best describes the domain of the tangent function $$y = \tan x$$?

$$x \neq \dfrac{\pi}{2} + k\pi$$ for all integers $$k$$

Q: A Ferris wheel rotates such that the height $$h (in $$meters) above the ground of a seat is modeled by $$h = 20 + 15 \sin \theta. $$What is the minimum value of $$\csc \theta$$ during the ride?

There is no minimum; $$\csc \theta is $$unbounded below.

Question 1

Which of the following best describes the domain of the tangent function $$y = 	an x$$?

Accepted Answer

$$x 
eq \dfrac{\pi}{2} + k\pi$$ for all integers $$k$$

Question 2

Where do the vertical asymptotes of the cotangent function $$y = \cot x$$ occur?

Accepted Answer

$$x = k\pi$$ for all integers $$k$$

Question 3

Which of the following statements about the secant function $$y = \sec x is $$true?

Accepted Answer

It has vertical asymptotes at the zeros of the cosine function.

Question 4

The period of the tangent function $$y = 	an x is$$:

Accepted Answer

$$\pi$$

Question 5

Which transformation occurs when graphing $$y = 3	an(\dfrac{1}{2}x) + 1$$ compared to $$y = 	an x$$?

Accepted Answer

Horizontal stretch by a factor of 2, vertical stretch by 3, up 1 unit

Question 6

The zeros of the tangent function $$y = 	an x$$ occur at:

Accepted Answer

$$x = k\pi$$ for all integers $$k$$

Question 7

Which of the following is the correct range of the secant function $$y = \sec x$$?

Accepted Answer

$$(-\infty, -1] \cup [1, \infty)$$

Question 8

A real-world application: A ladder leans against a wall, forming an angle $$	heta$$ with the ground. If the height $$h$$ the ladder reaches is given by $$h = 10 \sin 	heta$$, what is the value of $$\csc 	heta$$ when $$h = 5$$?

Accepted Answer

$$2$$

Question 9

Which of the following is NOT a property of the tangent function $$y = 	an x$$?

Accepted Answer

It is bounded above and below.

Question 10

If $$y = \sec x$$ and $$y = 	an x$$ intersect at $$x = 0.7749$$, what is the value of $$y at $$this intersection?

Accepted Answer

$$1.3996$$

Question 11

Which function has vertical asymptotes at $$x = k\pi$$ for all integers $$k$$?

Accepted Answer

$$y = \cot x$$

Question 12

The cosecant function $$y = \csc x is $$undefined whenever:

Accepted Answer

$$\sin x = 0$$

Question 13

Which of the following best describes the symmetry of the tangent function $$y = 	an x$$?

Accepted Answer

Odd symmetry about the origin

Question 14

A Ferris wheel rotates such that the height $$h (in $$meters) above the ground of a seat is modeled by $$h = 20 + 15 \sin 	heta. $$What is the minimum value of $$\csc 	heta$$ during the ride?

Accepted Answer

There is no minimum; $$\csc 	heta is $$unbounded below.

Question 15

Which of the following is true about the range of the tangent function $$y = 	an x$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 16

If $$y = \sec x is $$the reciprocal of $$y = \cos x$$, what happens to the graph of $$y = \sec x at $$the zeros of $$y = \cos x$$?

Accepted Answer

It has vertical asymptotes.

Question 17

Which of the following is the correct period of the cotangent function $$y = \cot x$$?

Accepted Answer

$$\pi$$

Question 18

A sound wave can be modeled by $$y = 2 	an(\omega t)$$, where $$\omega is $$the angular frequency. If the period of the wave is $$0.5$$ seconds, what is the value of $$\omega$$?

Accepted Answer

$$\omega = 2$$

Question 19

Which of the following is a zero of the cotangent function $$y = \cot x$$?

Accepted Answer

$$x = \dfrac{\pi}{2} + k\pi$$

Tangent, Cotangent, Secant, and Cosecant Functions in Precalculus

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