Domain, range, period and graphs

Maths · Trigonometric functions and their graphs. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Each trigonometric function is a function of a real number x (an angle in radians). The domain is the set of x for which the function is defined. The range is the set of values it can take. sin x and cos x are defined everywhere and stay between −1 and 1. The other four are undefined wherever their denominator is zero, and the graph has a vertical asymptote there.

All six functions repeat. The smallest positive number p with f(x + p) = f(x) is the period.

From the graph you should be able to read the zeros, the maximum and minimum values, and whether the function is even or odd.

Rules to remember

sin x and cos x: domain all real numbers, range −1 ≤ y ≤ 1
tan x and sec x are undefined at odd multiples of π/2 (x = ±π/2, ±3π/2, ...); cot x and cosec x are undefined at multiples of π (x = 0, ±π, ±2π, ...)
Range of tan x and cot x: all real numbers. Range of sec x and cosec x: y ≤ −1 or y ≥ 1
Period of sin, cos, sec and cosec is 2π; period of tan and cot is π
Period of sin kx or cos kx = 2π/k; period of tan kx = π/k (k > 0)
y = a sin x and y = a cos x (a > 0) have maximum a and minimum −a; sin x = 0 at x = nπ, cos x = 0 at x = π/2 + nπ
Even, f(−x) = f(x): cos x and sec x. Odd, f(−x) = −f(x): sin x, tan x, cot x and cosec x

Common mistake

For y = sin 3x students multiply and say the period is 6π. Divide instead: the period is 2π/3, because a larger number in front of x makes the graph repeat sooner.

Worked example

For y = 5 cos 2x, find the period, the maximum value and the smallest positive x at which the minimum occurs.

Period = 2π ÷ 2 = π.
cos 2x lies between −1 and 1, so the maximum of y is 5 × 1 = 5 and the minimum is 5 × (−1) = −5.
The minimum occurs when cos 2x = −1, so 2x = π and x = π/2.

Answer: Period π; maximum value 5; the minimum −5 first occurs at x = π/2.

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