Q: A ramp rises 10 meters over a horizontal distance of 20 meters. What angle $$\theta$$ does the ramp make with the ground? (Round to the nearest degree.)

$$\theta = \arctan\left(\frac{10}{20}\right) \approx 27^\circ$$

Q: Which transformation will shift the graph of $$y = \tan(x) to $$the right by $$\frac{\pi}{4}$$ units?

$$y = \tan\left(x - \frac{\pi}{4}\right)$$

Q: A ladder leans against a wall, making a $$75^\circ$$ angle with the ground. If the base of the ladder is 4 feet from the wall, how high up the wall does the ladder reach? (Round to the nearest tenth.)

$$4 \tan(75^\circ) \approx 15.0$$ feet

Q: A road rises at an angle of $$4.5^\circ$$ with the horizontal. What is the slope of the road?

$$\tan(4.5^\circ) \approx 0.079$$

Question 1

Which of the following best describes the period of the basic tangent function $$y = 	an(x)$$?

Accepted Answer

$$\pi$$

Question 2

What is the equation of the vertical asymptotes for the function $$y = 	an(x)$$?

Accepted Answer

$$x = \frac{\pi}{2} + n\pi$$, where $$n is $$any integer

Question 3

Given a line passing through the point $$(2, 3)$$ and making a $$60^\circ$$ angle with the positive $$x-$$axis, what is the equation of the line?

Accepted Answer

$$y - 3 = \sqrt{3}(x - 2)$$

Question 4

Which of the following is the correct formula for the cotangent function in terms of sine and cosine?

Accepted Answer

$$\cot(x) = \frac{\cos(x)}{\sin(x)}$$

Question 5

If $$y = 	an(2x)$$, what is the period of the function?

Accepted Answer

$$\frac{\pi}{2}$$

Question 6

Which of the following statements about the graph of $$y = \cot(x) is $$true?

Accepted Answer

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

Question 7

A ramp rises 10 meters over a horizontal distance of 20 meters. What angle $$	heta$$ does the ramp make with the ground? (Round to the nearest degree.)

Accepted Answer

$$	heta = \arctan\left(\frac{10}{20}ight) \approx 27^\circ$$

Question 8

Which transformation will shift the graph of $$y = 	an(x) to $$the right by $$\frac{\pi}{4}$$ units?

Accepted Answer

$$y = 	an\left(x - \frac{\pi}{4}ight)$$

Question 9

If $$y = -2	an(x)$$, which of the following describes the effect of the transformation on the basic tangent graph?

Accepted Answer

The graph is reflected over the $$x-$$axis and vertically stretched by a factor of 2.

Question 10

A ladder leans against a wall, making a $$75^\circ$$ angle with the ground. If the base of the ladder is 4 feet from the wall, how high up the wall does the ladder reach? (Round to the nearest tenth.)

Accepted Answer

$$4 	an(75^\circ) \approx 15.0$$ feet

Question 11

Which of the following points is a zero of the function $$y = 	an(x)$$?

Accepted Answer

$$x = 0$$

Question 12

Which of the following best describes the symmetry of the tangent function?

Accepted Answer

It is an odd function, symmetric about the origin.

Question 13

Given $$y = 	an(x)$$, which of the following is NOT a characteristic point used to sketch one period of the graph?

Accepted Answer

$$\left(\pi, 1\right)$$

Question 14

If a line passes through $$(1, -2)$$ and makes a $$45^\circ$$ angle with the $$x-$$axis, what is its equation?

Accepted Answer

$$y + 2 = 1(x - 1)$$

Question 15

Which of the following is the correct period for the function $$y = \cot(3x)$$?

Accepted Answer

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

Question 16

Which of the following is true about the zeros of $$y = \cot(x)$$?

Accepted Answer

They occur at $$x = \frac{\pi}{2} + n\pi$$, where $$n is $$any integer.

Question 17

A road rises at an angle of $$4.5^\circ$$ with the horizontal. What is the slope of the road?

Accepted Answer

$$	an(4.5^\circ) \approx 0.079$$

Graphs and Properties of Tangent and Cotangent Functions

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