AP Precalculus: The Tangent Function (Drill 1)

Question 1. What is the period of the function \( f(x) = \tan(x) \)?

• A) \( \dfrac{\pi}{2} \)
• B) \( \pi \) ✓
• C) \( 2\pi \)
• D) \( 4\pi \)

Explanation: Choice B is correct. Unlike sine and cosine, whose period is \( 2\pi \), the tangent function completes one full cycle over an interval of length \( \pi \). For example, \( \tan(x) \) passes through \( (0, 0) \), increases without bound as \( x \to \frac{\pi}{2}^- \), then continues from \( -\infty \) after the asymptote and returns to 0 at \( x = \pi \). Choice A is incorrect because \( \frac{\pi}{2} \) is the distance from the center of a cycle to the nearest vertical asymptote; it marks the asymptote location, not the full period. Choice C is incorrect because \( 2\pi \) is the period of \( \sin(x) \) and \( \cos(x) \), not \( \tan(x) \). Choice D is incorrect because \( 4\pi \) is not the period of any standard trigonometric function.

Question 2. Which of the following correctly describes the vertical asymptotes of \( f(x) = \tan(x) \)?

• A) \( x = n\pi \) for all integers \( n \)
• B) \( x = \dfrac{\pi}{2} + n\pi \) for all integers \( n \) ✓
• C) \( x = 2n\pi \) for all integers \( n \)
• D) \( x = \dfrac{\pi}{4} + n\pi \) for all integers \( n \)

Explanation: Choice B is correct. The tangent function is defined as \( \tan(x) = \sin(x)/\cos(x) \), so it is undefined wherever \( \cos(x) = 0 \). Cosine equals zero at \( x = \frac{\pi}{2} + n\pi \) for all integers \( n \), producing vertical asymptotes at those values. Note that this spacing of \( \pi \) between asymptotes, not \( 2\pi \), reflects the period of the tangent function. Choice A is incorrect because \( x = n\pi \) are the zeros of \( \tan(x) \) (where \( \sin(x) = 0 \)), not its asymptotes, a common swap error. Choice C is incorrect because \( x = 2n\pi \) are where \( \cos(x) = 1 \), giving \( \tan(x) = 0 \), not asymptotes. Choice D is incorrect because \( x = \frac{\pi}{4} + n\pi \) are where \( \tan(x) = \pm 1 \); these are defined values, not asymptotes.

Question 3. What is the period of \( g(x) = \tan(3x) \)?

• A) \( 3\pi \)
• B) \( \dfrac{2\pi}{3} \)
• C) \( \dfrac{\pi}{3} \) ✓
• D) \( \pi \)

Explanation: Choice C is correct. The period of \( \tan(Bx) \) is \( \pi / B \), not \( 2\pi / B \) as for sine and cosine. With \( B = 3 \), the period is \( \pi / 3 \). Choice A is incorrect because \( 3\pi \) results from multiplying \( \pi \) by \( B \) rather than dividing; a horizontal compression by factor 3 makes the period shorter, not longer. Choice B is incorrect because \( 2\pi / 3 \) uses the sine/cosine period formula \( 2\pi / B \) rather than the correct tangent formula \( \pi / B \), a common error when students apply the wrong base period. Choice D is incorrect because \( \pi \) is the period of \( \tan(x) \) with no coefficient; the factor of 3 compresses the period to one-third of that.

Question 4. On which of the following intervals is \( f(x) = \tan(x) \) continuous and increasing?

• A) \( \left(0,\, \pi\right) \)
• B) \( \left(-\dfrac{\pi}{2},\, \dfrac{\pi}{2}\right) \) ✓
• C) \( \left[-\dfrac{\pi}{2},\, \dfrac{\pi}{2}\right] \)
• D) \( \left(0,\, 2\pi\right) \)

Explanation: Choice B is correct. The tangent function is continuous and increasing on every open interval between consecutive vertical asymptotes. The principal such interval is \( \left(-\frac{\pi}{2},\, \frac{\pi}{2}\right) \); the endpoints are excluded because \( \tan(x) \) has vertical asymptotes there and is undefined. Choice A is incorrect because the interval \( \left(0,\, \pi\right) \) contains a vertical asymptote at \( x = \frac{\pi}{2} \), so \( \tan(x) \) is not continuous across the entire interval. Choice C is incorrect because the closed bracket notation \( \left[-\frac{\pi}{2},\, \frac{\pi}{2}\right] \) implies the endpoints are included, but \( \tan\!\left(\pm\frac{\pi}{2}\right) \) is undefined, the interval must be open. Choice D is incorrect because \( \left(0,\, 2\pi\right) \) contains vertical asymptotes at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \), so \( \tan(x) \) is neither continuous nor monotone on the full interval.

Question 5. A surveyor stands at point A and measures the angle of elevation θ to the top of a building. The horizontal distance from the surveyor to the base of the building is 80 feet. The height of the building is given by \( h = 80\tan(\theta) \). For which value of θ is this model undefined?

• A) \( \theta = 0 \)
• B) \( \theta = \dfrac{\pi}{4} \)
• C) \( \theta = \dfrac{\pi}{3} \)
• D) \( \theta = \dfrac{\pi}{2} \) ✓

Explanation: Choice D is correct. The model \( h = 80\tan(\theta) \) is undefined wherever \( \tan(\theta) \) is undefined. Since \( \tan(\theta) = \sin(\theta)/\cos(\theta) \), it is undefined when \( \cos(\theta) = 0 \), which occurs at \( \theta = \frac{\pi}{2} \). Geometrically, \( \theta = \frac{\pi}{2} \) corresponds to a vertical line of sight, which is not physically meaningful in this context, no finite horizontal distance can produce a 90° angle of elevation to a building top. Choice A is incorrect because \( \tan(0) = 0 \), giving \( h = 0 \), a defined value. Choice B is incorrect because \( \tan\!\left(\frac{\pi}{4}\right) = 1 \), giving \( h = 80 \) feet, also defined. Choice C is incorrect because \( \tan\!\left(\frac{\pi}{3}\right) = \sqrt{3} \), giving \( h = 80\sqrt{3} \approx 138.6 \) feet, which is defined.