AP Precalculus Quiz: The Tangent Function

What this quiz covers

This quiz focuses on The Tangent Function, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. At $x = k\pi$ for any integer $k$. (correct answer)
2. At $x = \frac{\pi}{2} + k\pi$ for any integer $k$.
3. At $x = \frac{\pi}{4} + \frac{k\pi}{2}$ for any integer $k$.
4. At $x = 2k\pi$ for any integer $k$.

Explanation: The zeros of $\tan(x) = \frac{\sin(x)}{\cos(x)}$ occur when the numerator, $\sin(x)$, is equal to 0, provided the denominator is not also 0. The function $\sin(x)$ is zero at all integer multiples of $\pi$. At these values, $\cos(x)$ is either 1 or -1, so the denominator is not zero.

Question 2

1. A vertical stretch by a factor of 2 and a reflection across the x-axis. (correct answer)
2. A vertical compression by a factor of 2 and a reflection across the y-axis.
3. A horizontal stretch by a factor of 2 and a reflection across the x-axis.
4. A vertical stretch by a factor of 2 and a translation 2 units down.

Explanation: For a function $y = af(x)$, the parameter $a$ causes a vertical stretch by a factor of $|a|$. If $a$ is negative, it also causes a reflection across the x-axis. In this case, $a=-2$, so there is a vertical stretch by a factor of 2 and a reflection across the x-axis.

Question 3

1. $x = 4\pi$
2. $x = 6\pi$ (correct answer)
3. $x = 3\pi$
4. $x = 10\pi$

Explanation: The period of $g(x) = \tan(\frac{1}{4}x)$ is $\frac{\pi}{|b|} = \frac{\pi}{1/4} = 4\pi$. The vertical asymptotes of a tangent function are separated by a distance equal to its period. If one asymptote is at $x=2\pi$, the next one for increasing $x$ will be at $x = 2\pi + \text{period} = 2\pi + 4\pi = 6\pi$.

Question 4

1. $f(-x) = -f(x)$ (correct answer)
2. $f(-x) = f(x)$
3. $f(x) = f(x+\pi)$
4. $f(x) = \frac{1}{f(x)}$

Explanation: The definition of an odd function is that $f(-x) = -f(x)$ for all $x$ in its domain. This corresponds to symmetry about the origin, which the tangent function possesses. Choice B defines an even function. Choice C is the definition of a periodic function with period $\pi$, which is true for tangent but is not the definition of an odd function.

Question 5

1. It translates the graph horizontally $\frac{\pi}{3}$ units to the left. (correct answer)
2. It translates the graph horizontally $\frac{\pi}{3}$ units to the right.
3. It translates the graph vertically $\frac{\pi}{3}$ units up.
4. It changes the period of the function to be $\frac{\pi}{3}$ units.

Explanation: For a function of the form $f(x+c)$, the graph is translated horizontally. If $c > 0$, the shift is to the left. Here, $c = \frac{\pi}{3}$, so the graph of $y=\tan(x)$ is shifted $\frac{\pi}{3}$ units to the left.

Question 6

1. Because both $\sin(x+\pi) = -\sin(x)$ and $\cos(x+\pi) = -\cos(x)$, their ratio remains unchanged. (correct answer)
2. Because the period of a ratio of functions is always half the period of the individual functions.
3. Because the tangent function has asymptotes which restrict the period to be smaller than $2\pi$.
4. Because the values of tangent in quadrant I are the reciprocals of values in quadrant IV.

Explanation: The period is the smallest positive value $P$ such that $f(x+P)=f(x)$. Let's test $P=\pi$. $\tan(x+\pi) = \frac{\sin(x+\pi)}{\cos(x+\pi)} = \frac{-\sin(x)}{-\cos(x)} = \frac{\sin(x)}{\cos(x)} = \tan(x)$. Since the function values repeat every $\pi$ units and this is the smallest such positive value, the period is $\pi$.

Question 7

1. The function is always increasing on the entire interval. (correct answer)
2. The function is always decreasing on the entire interval.
3. The function increases on $(\frac{\pi}{2}, \pi)$ and then decreases on $(\pi, \frac{3\pi}{2})$.
4. The function decreases on $(\frac{\pi}{2}, \pi)$ and then increases on $(\pi, \frac{3\pi}{2})$.

Explanation: The interval $(\frac{\pi}{2}, \frac{3\pi}{2})$ is one full period of the tangent function, between two consecutive vertical asymptotes. Throughout any such interval, the tangent function is strictly increasing.

Question 8

1. $\theta=\arctan\!\left(\dfrac{2}{3}\right)$ (correct answer)
2. $\theta=\arcsin\!\left(\dfrac{2}{3}\right)$
3. $\theta=\arccos\!\left(\dfrac{2}{3}\right)$
4. $\theta=\arctan\!\left(\dfrac{3}{2}\right)$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically using inverse tangent to find angles from known ratios. The tangent function is defined as the ratio of sine to cosine, and its inverse function arctan returns the angle whose tangent equals a given value. In this question, the architectural context with a 20m tower viewed from 30m away creates tan θ = 20/30 = 2/3. Choice A is correct because θ = arctan(2/3) properly uses the inverse tangent function to find the angle whose tangent equals 2/3. Choice D is incorrect because it inverts the fraction to 3/2, which would represent the angle if viewing from 20m away at a 30m tower. To help students: Emphasize that arctan 'undoes' the tangent function to recover angles. Practice setting up the opposite/adjacent ratio correctly before applying inverse tangent.

Question 9

1. $x=k\pi$
2. $x=\dfrac{k\pi}{2}$
3. $x=\dfrac{\pi}{2}+k\pi$ (correct answer)
4. $x=2k\pi$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically identifying where vertical asymptotes occur based on the function's definition. The tangent function is defined as sin θ/cos θ, which means it becomes undefined wherever cos θ = 0, creating vertical asymptotes at these points. In this question, students must identify all locations where cosine equals zero, which occurs at odd multiples of π/2. Choice C is correct because x = π/2 + kπ represents all odd multiples of π/2 (like π/2, 3π/2, 5π/2, etc.), which are precisely where cosine equals zero. Choice A is incorrect because x = kπ includes points like 0 and π where cosine equals 1 or -1, not zero. To help students: Draw the cosine graph and mark all zeros to visualize asymptote locations. Practice converting between different representations of periodic points, emphasizing that π/2 + kπ captures all odd multiples of π/2.

Question 10

1. $h=80\sin\theta$
2. $h=\dfrac{80}{\tan\theta}$
3. $h=80\tan\theta$ (correct answer)
4. $h=80\cos\theta$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically applying it to solve real-world height problems using angle of elevation. The tangent of an angle equals opposite over adjacent in a right triangle, so when viewing upward at angle θ from a horizontal distance, tan θ = height/horizontal distance. In this question, the drone is 80 m away horizontally and looks up at angle θ to see the lighthouse top, forming a right triangle. Choice C is correct because rearranging tan θ = h/80 gives h = 80 tan θ, properly using tangent to find height from horizontal distance and angle. Choice A is incorrect because h = 80 sin θ would require knowing the hypotenuse, not just the horizontal distance. To help students: Draw the scenario as a right triangle, labeling the 80 m as adjacent to θ and h as opposite. Practice setting up tangent ratios before solving, emphasizing that tangent relates the two legs of a right triangle, not involving the hypotenuse.

Question 11

1. $x=\dfrac{\pi}{2}+k\pi$
2. $x=\dfrac{\pi}{4}+k\pi$
3. $x=k\pi$ (correct answer)
4. $x=\dfrac{k\pi}{2}$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically locating its zeros based on the quotient definition. Since tan x = sin x/cos x, the function equals zero when the numerator sin x = 0 and the denominator cos x ≠ 0, which occurs at integer multiples of π. In this question, students must identify where sin x = 0 while ensuring cos x ≠ 0 to avoid undefined points. Choice C is correct because x = kπ represents all integer multiples of π (0, ±π, ±2π, etc.), where sine equals zero and cosine equals ±1. Choice A is incorrect because x = π/2 + kπ represents the asymptotes where cos x = 0, not the zeros of tangent. To help students: Graph y = sin x, y = cos x, and y = tan x together to visualize where tangent crosses the x-axis. Emphasize that zeros occur where the numerator is zero but the denominator isn't, distinguishing zeros from undefined points.

Question 12

1. Opposite is 4 when adjacent is 3.
2. Opposite is 3 when adjacent is 4. (correct answer)
3. Hypotenuse is 3 when adjacent is 4.
4. Adjacent is 3 when hypotenuse is 4.

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically interpreting tangent ratios in practical surveying contexts. When tan θ = 3/4, this means the opposite side is 3 units for every 4 units of the adjacent side, following the definition tan θ = opposite/adjacent. In this question, students must correctly interpret the given ratio tan θ = 3/4 in terms of the triangle's sides. Choice B is correct because it accurately states that the opposite side is 3 when the adjacent side is 4, matching the given tangent value. Choice A is incorrect because it reverses the ratio, stating opposite is 4 when adjacent is 3, which would give tan θ = 4/3, not 3/4. To help students: Practice writing tangent ratios in fraction form and identifying which number represents opposite versus adjacent. Use proportional reasoning to scale triangles while maintaining the same angle, showing that tan θ = 3/4 could also mean opposite = 6 when adjacent = 8.

Question 13

1. Tangent equals $\dfrac{\text{run}}{\text{rise}}$.
2. Tangent equals $\dfrac{\text{rise}}{\text{run}}$. (correct answer)
3. Tangent equals $\dfrac{\text{hypotenuse}}{\text{run}}$.
4. Tangent equals $\dfrac{\text{rise}}{\text{hypotenuse}}$.

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically its direct relationship to slope in practical applications. The tangent of an angle in a right triangle equals the ratio of the opposite side to the adjacent side, which corresponds exactly to rise over run in slope calculations. In this question, the road grade context connects mathematical concepts to real-world engineering applications. Choice B is correct because tangent equals rise/run, matching the standard definition of slope in coordinate geometry. Choice A is incorrect because it inverts the ratio to run/rise, which would give the cotangent rather than the tangent. To help students: Draw right triangles with various angles and calculate both tan θ and slope to see they're identical. Use physical models like ramps or stairs to demonstrate that steeper angles have larger tangent values, just like steeper lines have larger slopes.

Question 14

1. $\theta=\dfrac{\pi}{3}$
2. $\theta=\dfrac{\pi}{2}$ (correct answer)
3. $\theta=\pi$
4. $\theta=\dfrac{3\pi}{2}$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically identifying where the function is undefined due to division by zero. The tangent function equals sin θ/cos θ, so it's undefined whenever the denominator cos θ equals zero, which first occurs at θ = π/2 in the standard domain. In this question, students must recognize that cos(π/2) = 0, making tan(π/2) undefined. Choice B is correct because at θ = π/2, cosine equals zero while sine equals one, creating a division by zero situation. Choice A is incorrect because cos(π/3) = 1/2, not zero, so tan(π/3) = √3 is well-defined. To help students: Create a table of special angles showing sin θ, cos θ, and tan θ values to identify patterns. Emphasize checking the denominator (cosine) for zeros when determining where tangent is undefined, and connect this to vertical asymptotes on the graph.

Question 15

1. $x=k\pi$
2. $x=\dfrac{\pi}{2}+k\pi$ (correct answer)
3. $x=2k\pi$
4. $x=\dfrac{\pi}{2}+2k\pi$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically identifying where vertical asymptotes occur. The tangent function is defined as the ratio of sine to cosine, and vertical asymptotes occur where the denominator (cosine) equals zero. In this question, students must determine when cos(x) = 0, which happens at odd multiples of π/2. Choice B is correct because x = π/2 + kπ represents all odd multiples of π/2 where k is any integer, precisely where cosine equals zero. Choice A is incorrect because it represents multiples of π where cosine alternates between 1 and -1, not zero. To help students: Emphasize that vertical asymptotes occur when denominators equal zero. Practice identifying zeros of cosine by visualizing the unit circle or cosine graph.

Question 16

1. tan $\theta=\dfrac{\text{rise}}{\text{run}}$ (correct answer)
2. sin $\theta=\dfrac{\text{rise}}{\text{run}}$
3. cos $\theta=\dfrac{\text{rise}}{\text{run}}$
4. tan $\theta=\dfrac{\text{run}}{\text{rise}}$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically its relationship to slope in right triangles. The tangent function is defined as the ratio of sine to cosine, which in a right triangle context equals opposite over adjacent, or rise over run. In this question, the ramp scenario with 3m rise and 4m run provides a concrete application of tangent as slope. Choice A is correct because tan θ = rise/run directly matches the slope formula, making tangent the natural trigonometric function for calculating slopes and angles of inclination. Choice D is incorrect because it inverts the ratio, giving run/rise which would be cotangent, not tangent. To help students: Connect tangent to the slope formula m = rise/run from algebra. Use physical examples like ramps and ladders to reinforce the opposite/adjacent interpretation.

Question 17

1. $\pi$ (correct answer)
2. $2\pi$
3. $\dfrac{\pi}{2}$
4. $\dfrac{3\pi}{2}$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically its fundamental period and how it differs from sine and cosine functions. The tangent function is periodic with period π, not 2π like sine and cosine, because tan(x + π) = tan(x) for all x in the domain. In this question, the navigation context emphasizes the practical importance of understanding periodicity for cyclic phenomena. Choice A is correct because the tangent function completes one full cycle every π radians, visible in its graph between consecutive asymptotes. Choice B is incorrect because it confuses tangent's period with that of sine and cosine, which have period 2π. To help students: Graph y = tan x alongside y = sin x and y = cos x to compare periods visually. Emphasize that tangent's period is half that of sine and cosine because both sin(x + π) and cos(x + π) change sign, making their ratio unchanged.

Question 18

1. $2\pi$
2. $\pi$ (correct answer)
3. $\dfrac{\pi}{2}$
4. $4\pi$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically its fundamental period. The tangent function is defined as the ratio of sine to cosine, and unlike sine and cosine which have period 2π, tangent has a period of π. In this question, the angle-of-elevation context reinforces the practical application while testing period knowledge. Choice B is correct because the tangent function repeats its pattern every π radians, as tan(x + π) = tan(x) for all x in the domain. Choice A is incorrect because it confuses tangent's period with that of sine and cosine, which have period 2π. To help students: Draw multiple periods of the tangent graph to visualize the π-interval repetition. Compare graphs of sin, cos, and tan side-by-side to distinguish their different periods.

Question 19

1. Slope $m=\sin\theta$
2. Slope $m=\cos\theta$
3. Slope $m=\tan\theta$ (correct answer)
4. Slope $m=\cot\theta$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically its direct relationship to the slope of a line. The tangent function is defined as the ratio of sine to cosine, and when a line makes angle θ with the positive x-axis, its slope equals tan θ. In this question, students must connect the geometric interpretation of tangent to the algebraic concept of slope. Choice C is correct because the slope m of a line making angle θ with the positive x-axis is precisely tan θ, linking trigonometry to coordinate geometry. Choice A is incorrect because sin θ would give the y-coordinate on the unit circle, not the slope of the line. To help students: Draw lines at various angles and calculate both their slopes and tangents to see they match. Emphasize that tangent measures 'steepness' just like slope does in algebra.

Question 20

1. $\theta=\sin^{-1}\!\left(\dfrac{1.2}{6.0}\right)$
2. $\theta=\tan^{-1}\!\left(\dfrac{1.2}{6.0}\right)$ (correct answer)
3. $\theta=\cos^{-1}\!\left(\dfrac{1.2}{6.0}\right)$
4. $\theta=\tan^{-1}\!\left(\dfrac{6.0}{1.2}\right)$

Explanation: This question tests AP Precalculus understanding of the tangent function's properties, specifically its application to real-world angle calculations using inverse functions. The tangent function is defined as the ratio of sine to cosine, and when given a tangent value, we use the inverse tangent function to find the angle. In this question, the ramp scenario provides a rise of 1.2 m and a run of 6.0 m, establishing that tan θ = 1.2/6.0. Choice B is correct because it properly applies the inverse tangent function to the given ratio: θ = tan⁻¹(1.2/6.0). Choice D is incorrect because it inverts the ratio to 6.0/1.2, which would give the cotangent rather than the tangent of the angle. To help students: Emphasize that tangent equals rise over run in right triangle applications. Practice setting up ratios correctly before applying inverse functions, and use diagrams to visualize which sides represent rise and run.