JEE Mains Maths · Trigonometric Equations

Exponential, Bounded and Graphical Equations

Equations that are not a polynomial in one ratio: powers with a trigonometric exponent, sides that meet only at their bounds, a line against the tangent graph, and moduli.

All Trigonometric Equations notes JEE Mains Maths strategy

Why this matters

Thirteen PYQs, ten of them multiple choice. Four put sin², cos², tan² or sec² in an exponent and become an equation in one power; nine have no algebraic route, so they compare bounds on the two sides, draw a line against the tangent graph, use a function that only increases, or split a modulus by sign. Two ideas cover the page.

Concept 1 of 2: Powers with sin² and cos² in the exponent

a^{\sin^2x}+a^{\cos^2x}

the exponents add to 1, so with

t=a^{\sin^2x}

the second term is

\frac at

. The equation becomes a quadratic in

t

. Each root gives a value of

\sin^2x

, which must lie in

[0,1]

. The same works for

\tan^2x

and

\sec^2x

, which differ by 1.

Definition

$a^{\cos^2x}=\frac{a}{a^{\sin^2x}}$ and $a^{\sec^2x}=a\cdot a^{\tan^2x}$ .
For $a>1$ , $t=a^{\sin^2x}$ lies in $[1,a]$ ; keep only roots in that range.
Then $\sin^2x=\log_at$ ; count the angles as usual.

Substitution

t=a^{\sin^2x}:\qquad a^{\sin^2x}+a^{\cos^2x}=t+\frac at

Worked example

How many solutions has

4^{\sin^2x}+4^{\cos^2x}=5

[0,2\pi]

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 10 April 2023 · Q165Moderate

Example 1 · Trigonometric Equations · Exponential, Bounded and Graphical Equations

Let

S =\left\{ x \in\left( -\frac{\pi}{2},\frac{\pi}{2} \right):9^{1 -\tan^{2}x}+9^{\tan^{2}x}= 10 \right\}

and

\beta =\sum_{x \in S} \tan^{2}\left( \frac{x}{3} \right)

, then

\frac{1}{6}(\beta - 14)^{2}

is equal to

A root in t may be out of range

For

a>1

t=a^{\sin^2x}

lies in

[1,a]

. A root of the quadratic outside that range gives no angle, just as

\sin x=2

gives none.

Drill 3 more on powers with sin² and cos² in the exponent

Concept 2 of 2: Bounds, graphs and moduli

Some equations have no algebraic route. If one side is at most 2 and the other at least 2, both must equal 2 at the same point. If a line meets

\tan x

, count one crossing per full branch and check the part-branches at the ends. If a function only increases, it has at most one root. A modulus is split by sign, and each root is kept only if it has the sign its case assumed.

Definition

Bounds: $2\cos u\le2$ and $a^x+a^{-x}\ge2$ , with equality only at $x=0$ .
$\sin^7x+\cos^7x\le\sin^2x+\cos^2x=1$ , with equality only when one of $\sin x,\cos x$ is 1 and the other 0.
A decreasing line meets each full branch of $\tan x$ exactly once.
$f'>0$ throughout: at most one root, and exactly one if $f$ changes sign.
$|u|=v$ : solve $u=v$ where $u\ge0$ and $-u=v$ where $u<0$ ; keep roots that fit their case.

Forced equality

f(x)\le m\le g(x)\ \text{for all}\ x:\quad f(x)=g(x)\ \text{exactly when}\ f(x)=g(x)=m

Worked example

How many real

x

satisfy

2\sin x=3^x+3^{-x}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 3 Apr 2025 · Q60Moderate

Example 2 · Trigonometric Equations · Exponential, Bounded and Graphical Equations

The number of solutions of the equation

2x+ 3\tan x=\pi,x\in\left\lbrack - 2\pi,2\pi \right\rbrack-

\left\{ \pm\frac{\pi}{2}, \pm\frac{3\pi}{2} \right\}

is.

Check the part-branches at the ends

A full branch of

\tan x

always meets a decreasing line once. A part-branch at the edge of the interval may or may not: compare the line's values there with the range that

\tan x

covers on that piece.

Drill 8 more on bounds, graphs and moduli

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Powers with sin² and cos² in the exponent
Substitution
$t=a^{\sin^2x}:\qquad a^{\sin^2x}+a^{\cos^2x}=t+\frac at$
Bounds, graphs and moduli
Forced equality
$f(x)\le m\le g(x)\ \text{for all}\ x:\quad f(x)=g(x)\ \text{exactly when}\ f(x)=g(x)=m$

Watch out for (2)

A root in t may be out of range
→ Powers with sin² and cos² in the exponent
Check the part-branches at the ends
→ Bounds, graphs and moduli

Test yourself on Trigonometric Equations

15 past JEE Mains questions from this chapter, timed at 36 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Trigonometric Equations chapter test Drill all 13 questions