Questions: Graphing Tangent and Reciprocal Trigonometric Functions

Question 1

The graph of y = csc(x) has vertical asymptotes at which x-values?

• A. x = π/2 + nπ, where n is any integer
• B. x = nπ, where n is any integer
• C. x = π/4 + nπ/2, where n is any integer
• D. x = 2nπ, where n is any integer

Answer: (B)

Since csc(x) = 1/sin(x), vertical asymptotes occur wherever sin(x) = 0, which is at x = nπ. Option A (x = π/2 + nπ) is where cos(x) = 0, making those the asymptotes of sec(x) — a common confusion between the two reciprocal functions.

Question 2

A student claims y = tan(x) has amplitude 1 because it passes through (π/4, 1) and (−π/4, −1). What is wrong with this claim?

• A. The student used the wrong points — tan(x) reaches its maximum at x = π/2
• B. Amplitude is only defined for functions with a maximum and minimum; tan(x) is unbounded and has no amplitude
• C. The amplitude of tan(x) is actually π, equal to its period
• D. Nothing is wrong — the amplitude of tan(x) is 1

Answer: (B)

Amplitude is defined as half the distance from minimum to maximum value. Since tan(x) increases without bound toward +∞ near each asymptote and decreases toward −∞, it has no maximum or minimum — and therefore no amplitude. The values 1 and −1 are just particular outputs at specific inputs, not extremes of the function.

Question 3

The graph of y = sec(x) and the graph of y = cos(x) touch (intersect) at every point where cos(x) = ±1.

• T. True
• F. False

Answer: True

Since sec(x) = 1/cos(x), when cos(x) = 1 we get sec(x) = 1, and when cos(x) = −1 we get sec(x) = −1. These are the only points where a function and its reciprocal share the same value. Between these contact points, sec(x) bows outward (away from the x-axis) while cos(x) curves inward — they never cross between the extremes.

Question 4

The period of y = tan(x) is 2π, the same as the period of y = sin(x) and y = cos(x).

• T. True
• F. False

Answer: False

The period of y = tan(x) is π, not 2π. Because tan(x) = sin(x)/cos(x), each full cycle of the tangent — sweeping from −∞ to +∞ between consecutive asymptotes — spans only π units. You can verify this: tan(0) = 0 and tan(π) = 0 with the same behavior on each side, so the pattern repeats every π. Confusing the period of tangent with the period of sine and cosine is one of the most common errors when first studying these graphs.

Question 5

Explain why the zeros of y = sin(x) become vertical asymptotes on the graph of y = csc(x), using the relationship between a function and its reciprocal.

Model answer: Since csc(x) = 1/sin(x), the value of csc(x) is defined as long as sin(x) ≠ 0. When sin(x) approaches 0, 1/sin(x) grows without bound toward ±∞, producing a vertical asymptote. There is no way to assign a finite value to 1/0, so csc(x) is undefined at those points. More generally: wherever any function equals zero, its reciprocal is undefined, which means zeros of the parent always become asymptotes of the reciprocal function.

This principle — zeros become asymptotes, extremes become contact points — is the key to sketching all four reciprocal trig functions (csc, sec) and cotangent. Drawing the parent curve lightly first, then identifying its zeros and extremes, gives you all the structural information needed for the reciprocal graph.