MathematicsRelations and FunctionsJEE

Relations and Functions · Part 13 of 20

Domain and Range of Trigonometric Expressions

Eight JEE-style problems on domains, ranges and solution counts of trigonometric expressions, all solved with one method: name the trig part t, find where t lives, then work on t.

Builds on: Part 12 · Trigonometric Functions: Graphs and Ranges.

Domain and Range of Trigonometric Expressions

Video coming soon

Method

Substitute t

sin²x − sin x + 1 = (t − ½)² + ¾ with t ∈ [−1, 1]: range [3/4, 3]. 1/(2 cos x − 1): u ∈ [−3, 1], u ≠ 0, so the range is (−∞, −1/3] ∪ [1, ∞).

Worked substitutions. — Vertex inside the interval gives the minimum.

cos x and |cos x| compared. — Open intervals: the tan restriction removes the ends.

Identical functions

cos x vs 1/√(1 + tan²x)

√(1 + tan²x) = |sec x|, so the second function is |cos x|, defined where cos x ≠ 0. They agree where cos x > 0: ∪(2nπ − π/2, 2nπ + π/2).

Counting solutions

sin x = x/10

Only |x| ≤ 10 matters. One crossing on (0, π), two on (2π, 3π), mirror images, and x = 0: 7 solutions.

sin x and x/10 with crossings marked. — Shaded: |x| > 10, no solutions.

Worked examples

From the video

1. Domain of 1/(1 + 2 sin x).

R − {nπ + (−1)ⁿ 7π/6}.

2. Domain of √(cos(sin x)).

3. Range of sin²x − sin x + 1.

[3/4, 3].

4. Range of 1/(2 cos x − 1).

(−∞, −1/3] ∪ [1, ∞).

5. For which x are cos x and 1/√(1 + tan²x) identical?

∪(2nπ − π/2, 2nπ + π/2).

6. f is defined on [0, 1]. Domain of f(tan x).

∪[nπ, nπ + π/4].

7. Range of 3 sin(√(π²/16 − x²)).

[0, 3/√2].

8. Number of solutions of sin x = x/10.

JEE-style question

Your turn

The number of real solutions of sin x = x/5 is:

(a)1

(b)3

(c)5

(d)7

Show the answer and the traps

Only |x| ≤ 5 matters. One crossing on (0, π), none beyond, since 2π > 5. With the mirror and x = 0: 3. Option (b).

1 counts only x = 0. 5 adds a hump that starts at 2π ≈ 6.3, beyond 5.

7 is the answer for x/10, not x/5.

Watch out

Common mistakes

Simplifying √(1 + tan²x) to sec xIt is |sec x|, and tan x must exist.

Plugging only the ends of t ∈ [−1, 1]Check whether the vertex lies inside.

Counting crossings beyond |x| ≤ k for sin x = x/k|sin x| ≤ 1 limits the search.

Practice

Try these

1. Domain of √(sin x).

∪[2nπ, (2n + 1)π].

2. Range of cos²x + cos x + 1.

[3/4, 3].

3. Range of 1/(3 + sin x).

[1/4, 1/2].

4. Domain of log(cos x).

∪(2nπ − π/2, 2nπ + π/2).