MathematicsRelations and FunctionsJEE

Relations and Functions · Part 12 of 20

Trigonometric Functions: Graphs and Ranges

The six trigonometric functions read from their graphs: domain, range, period, parity and general solutions, plus the range of a cos x + b sin x.

Builds on: Part 10 · Domain and Range of Algebraic Functions.

Video coming soon

Graphs

The six trig graphs

sin and cos are smooth waves in [−1, 1]; tan and cot repeat every π with vertical asymptotes; sec and cosec never take values strictly between −1 and 1.

Six small graphs. — Dashed lines are asymptotes.

sin and cos

Where sin x ≥ 0

sin x ≥ 0 on [2nπ, (2n + 1)π]; cos x ≥ 0 on [2nπ − π/2, 2nπ + π/2]. Both have range [−1, 1] and period 2π.

Summary

Domain, range, period, parity

tan and cot have period π; the other four 2π. cos and sec are even; sin, tan, cot and cosec are odd. tan and sec break at odd multiples of π/2, cot and cosec at multiples of π.

Derivation

$a cos x + b sin x$

With $r = a^{2} + b^{2}$ , sin α = a/r, cos α = b/r: $a cos x + b sin x = r sin (x + α)$ , so the range is [−r, r]. 3 cos x + 4 sin x ∈ [−5, 5].

Summary

Key formulas

sin x \geq 0 ⟺ x \in ⋃ [2 nπ, (2 n + 1) π]

Transcendental functions and the six trig graphs

cos x \geq 0 ⟺ x \in ⋃ [2 nπ - \frac{π}{2}, 2 nπ + \frac{π}{2}]

Transcendental functions and the six trig graphs

a cos x + b sin x = a^{2} + b^{2} sin (x + α)

Range of a cos x + b sin x

range [- a^{2} + b^{2}, a^{2} + b^{2}]

Range of a cos x + b sin x

Worked examples

From the video

1. General solution of sin x = sin α.

x = nπ + (−1)ⁿα.

2. Range of 3 cos x + 4 sin x.

[−5, 5].

JEE-style question

Your turn

The range of 5 sin x − 12 cos x + 7 is:

(a)[−6, 20]

(b)[−13, 13]

(c)[−10, 24]

(d)[2, 12]

Show the answer and the traps

√(25 + 144) = 13, so 5 sin x − 12 cos x ∈ [−13, 13], and the whole thing is in [−6, 20]. Option (a).

(b) forgets the + 7. (c) uses 5 + 12 = 17 instead of √(5² + 12²).

(d) uses only the 5 sin x part: 7 ± 5.

Watch out

Common mistakes

Using |a| + |b| for the range of a cos x + b sin xIt is √(a² + b²).

Giving tan x a period of 2πtan and cot repeat every π.

Putting values between −1 and 1 in the range of sec x|sec x| ≥ 1.

Practice

Try these

1. Range of 3 sin x + 4 cos x − 2.

[−7, 3].

2. Domain of sec x + cosec x.

R − {nπ/2}.

3. General solution of cos x = 1/2.

x = 2nπ ± π/3.

4. On [0, 2π], where are sin x and cos x both ≥ 0?

[0, π/2].