How much of JEE Main comes from Trigonometry?

JEEGenie estimates about 5% of JEE Main marks come from Trigonometry, based on recent papers. The share varies from year to year.

Which formulas matter most in Trigonometry?

The Trigonometry sheet on this page has 23 formula entries, starting with Angle Measurement: Degrees & Radians, Trigonometric Ratios: Definitions, Pythagorean Identities, Sum and Difference Formulas, Double Angle Formulas. Each one comes with when to use it and the trap to avoid.

What topics does Trigonometry cover for JEE?

Trigonometry covers Trigonometric Ratios and Identities, Compound and Multiple Angles, Trigonometric Equations, Inverse Trigonometric Functions, Properties of Triangles, Heights and Distances.

How should I revise Trigonometry before the exam?

Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Trigonometry questions on JEEGenie.

How many previous-year questions are there on Trigonometry?

JEEGenie has 488 previous-year questions on Trigonometry, and this page shows a few with worked explanations.

Is practice on Trigonometry free on JEEGenie?

Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.

Trigonometry for JEE Main & Advanced — formulas, what is asked, and traps

JEEGenie estimates about 5% of JEE Main marks come from Trigonometry. This page covers 23 formula entries, each with when to use it, what JEE asks and the traps. A common slip: Forgetting to convert degrees to radians when using calculus or arc length formulas.

Key points to know

Angle Measurement: Degrees & Radians. An angle is the measure of rotation. Radian measure is the ratio of arc length to radius, with $1 radian = \frac{180^{\circ}}{π}$ .
Trigonometric Ratios: Definitions. For a right triangle with angle $θ$ , the six ratios are: $\sin θ = \frac{opposite}{hypotenuse}$ , $\cos θ = \frac{adjacent}{hypotenuse}$ , $\tan θ = \frac{\sin θ}{\cos θ}$ , and their reciprocals.
Trigonometric Ratios of Standard Angles. Memorize exact values for $0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}$ and their radian equivalents.
Unit Circle and Sign of Trigonometric Functions. On the unit circle ( $x^{2} + y^{2} = 1$ ), $\cos θ = x$ , $\sin θ = y$ . Signs depend on quadrant: ASTC (All positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4).
Trigonometric Ratios of Allied Angles. Angles like $90^{\circ} \pm θ$ , $180^{\circ} \pm θ$ have simple relationships: e.g., $\sin (90^{\circ} - θ) = \cos θ$ , $\cos (180^{\circ} - θ) = - \cos θ$ .
Periodicity and Symmetry of Trigonometric Functions. Sine and cosine have period $2 π$ , tangent has period $π$ . Even/odd: $\cos$ is even, $\sin$ and $\tan$ are odd.
Trigonometric Substitutions in Integration. Use $x = a \sin θ$ for $\sqrt{a^{2} - x^{2}}$ , $x = a \tan θ$ for $\sqrt{a^{2} + x^{2}}$ , $x = a \sec θ$ for $\sqrt{x^{2} - a^{2}}$ .
Heights and Distances: Angle of Elevation/Depression. Angle of elevation is measured upward from horizontal; angle of depression is measured downward. Use trig ratios to find heights or distances.

Formula sheet

Name	Formula	When to use
Angle Measurement: Degrees & Radians	$1^{\circ} = \frac{π}{180} rad$ and $1 rad = \frac{180^{\circ}}{π}$	Whenever angles are given in degrees or radians, especially in arc length, sector area, or trigonometric function evaluations.
Trigonometric Ratios: Definitions	$\csc θ = \frac{1}{\sin θ}$ , $\sec θ = \frac{1}{\cos θ}$ , $\cot θ = \frac{1}{\tan θ}$	For any problem involving right triangles or defining trigonometric functions in terms of sides.
Pythagorean Identities	$\sin^{2} θ + \cos^{2} θ = 1$ , $1 + \tan^{2} θ = \sec^{2} θ$ , $1 + \cot^{2} θ = \csc^{2} θ$	To simplify expressions, prove identities, or solve equations involving squares of trig functions.
Sum and Difference Formulas	$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$ , $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$ , $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$	To find exact values of trig functions for angles like $15^{\circ}$ , $75^{\circ}$ , or to simplify expressions.
Double Angle Formulas	$\sin 2 θ = 2 \sin θ \cos θ$ , $\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$ , $\tan 2 θ = \frac{2 \tan θ}{1 - \tan^{2} θ}$	To simplify expressions with $2 θ$ , solve equations, or integrate trig functions.
Half Angle Formulas	$\sin \frac{θ}{2} = \pm \sqrt{\frac{1 - \cos θ}{2}}$ , $\cos \frac{θ}{2} = \pm \sqrt{\frac{1 + \cos θ}{2}}$ , $\tan \frac{θ}{2} = \pm \sqrt{\frac{1 - \cos θ}{1 + \cos θ}} = \frac{\sin θ}{1 + \cos θ} = \frac{1 - \cos θ}{\sin θ}$	To find exact values for angles like ${22.5}^{\circ}$ , or to simplify integrals and equations.
Product-to-Sum and Sum-to-Product Formulas	$\sin A \cos B = \frac{1}{2} [\sin (A + B) + \sin (A - B)]$ , $\cos A \cos B = \frac{1}{2} [\cos (A + B) + \cos (A - B)]$ , $\sin A \sin B = \frac{1}{2} [\cos (A - B) - \cos (A + B)]$ , $\sin C + \sin D = 2 \sin \frac{C + D}{2} \cos \frac{C - D}{2}$ , $\cos C + \cos D = 2 \cos \frac{C + D}{2} \cos \frac{C - D}{2}$ , $\cos C - \cos D = - 2 \sin \frac{C + D}{2} \sin \frac{C - D}{2}$	To simplify products of trig functions, solve equations, or integrate products.
Solving Trigonometric Equations	General solutions: $\sin θ = \sin α \Rightarrow θ = n π + (- 1)^{n} α$ , $\cos θ = \cos α \Rightarrow θ = 2 n π \pm α$ , $\tan θ = \tan α \Rightarrow θ = n π + α$ , where $n \in Z$ .	When given an equation involving trigonometric functions, find all angles satisfying it.
Trigonometric Ratios of Standard Angles	$\sin 0 = 0$ , $\sin 30^{\circ} = \frac{1}{2}$ , $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ , $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$ , $\sin 90^{\circ} = 1$ ; similarly for $\cos$ and $\tan$ .	For quick evaluation of trig functions at standard angles, especially in geometry and calculus.
Unit Circle and Sign of Trigonometric Functions	Quadrant signs: Q1: all +, Q2: $\sin$ +, Q3: $\tan$ +, Q4: $\cos$ +	To determine sign of trig functions for any angle, especially in solving equations or evaluating expressions.
Graphs of Trigonometric Functions	$y = \sin x$ : domain $R$ , range $[- 1, 1]$ , period $2 π$ ; $y = \tan x$ : domain $x \neq \frac{π}{2} + n π$ , range $R$ , period $π$ .	To understand behavior of trig functions, solve inequalities, or find number of solutions.
Inverse Trigonometric Functions: Principal Values	$\sin^{- 1} (\sin x) = x$ only if $x \in [- \frac{π}{2}, \frac{π}{2}]$ ; otherwise use periodicity.	To find angles from given trig values, especially in calculus and solving equations.
Trigonometric Identities: Proving and Simplifying	Example: $\frac{\sin θ}{1 + \cos θ} + \frac{1 + \cos θ}{\sin θ} = 2 \csc θ$	To prove trigonometric identities or simplify complex trig expressions.
Trigonometric Ratios of Allied Angles	$\sin (π - θ) = \sin θ$ , $\cos (π - θ) = - \cos θ$ , $\tan (π - θ) = - \tan θ$	To evaluate trig functions of angles like $120^{\circ}$ , $135^{\circ}$ , etc., quickly.
Maximum and Minimum Values of Trigonometric Expressions	$a \sin θ + b \cos θ = R \sin (θ + ϕ)$ where $R = \sqrt{a^{2} + b^{2}}$ , $\cos ϕ = a / R, \sin ϕ = b / R$	To find max/min of expressions like $3 \sin x + 4 \cos x$ or to solve inequalities.
Sine and Cosine Laws for Triangles	Sine law: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2 R$ ; Cosine law: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$	To solve triangles (find unknown sides/angles) given some sides and angles.
Area of a Triangle: Trigonometric Formula	$Area = \frac{1}{2} a b \sin C = \frac{1}{2} b c \sin A = \frac{1}{2} c a \sin B$	To find area when two sides and the included angle are known.
Periodicity and Symmetry of Trigonometric Functions	$\sin (- θ) = - \sin θ$ , $\cos (- θ) = \cos θ$ , $\tan (- θ) = - \tan θ$	To simplify expressions with negative angles or to find general solutions.
Solving Trigonometric Inequalities	Example: $\sin x > \frac{1}{2}$ gives $x \in (\frac{π}{6}, \frac{5 π}{6}) + 2 n π$	When given an inequality involving trig functions, find the range of angles.
Trigonometric Substitutions in Integration	$\sqrt{a^{2} - x^{2}} = a \cos θ$ when $x = a \sin θ$	To integrate expressions involving square roots of quadratic forms.
Trigonometric Equations: Using Auxiliary Angle	$a \sin θ + b \cos θ = \sqrt{a^{2} + b^{2}} \sin (θ + ϕ)$ with $\cos ϕ = a / R, \sin ϕ = b / R$	To solve equations of the form $a \sin θ + b \cos θ = c$ .
Heights and Distances: Angle of Elevation/Depression	$\tan θ = \frac{height}{distance}$	In word problems involving towers, buildings, or objects at a distance.
Trigonometric Series: Summation of Sine/Cosine Series	Sum of $\sin$ series: $\frac{\sin (n β / 2)}{\sin (β / 2)} \sin (α + (n - 1) β / 2)$	To find sum of trigonometric series in arithmetic progression.

Open the Trigonometry formula page

What JEE asks

Angle Measurement: Degrees & Radians. JEE often tests conversion in problems involving trigonometric equations or geometry, e.g., finding arc length or sector area.
Trigonometric Ratios: Definitions. JEE uses these definitions to derive identities and solve triangle problems, often in conjunction with Pythagorean theorem.
Pythagorean Identities. JEE frequently uses these to transform expressions, e.g., converting $\sin^{2} θ$ to $1 - \cos^{2} θ$ .
Sum and Difference Formulas. JEE often tests these in proving identities or solving equations, e.g., $\sin (45^{\circ} + x)$ .
Double Angle Formulas. JEE uses these to reduce powers, e.g., $\sin^{2} θ = \frac{1 - \cos 2 θ}{2}$ .
Half Angle Formulas. JEE tests these in problems requiring exact trigonometric values or in integration.
Product-to-Sum and Sum-to-Product Formulas. JEE often uses these to solve trigonometric equations like $\sin 5 x + \sin 3 x = 0$ .
Solving Trigonometric Equations. JEE often asks for solutions in a specific interval, e.g., $[0, 2 π]$ , or general solutions.
Trigonometric Ratios of Standard Angles. JEE tests these in problems like finding exact values or simplifying expressions.
Unit Circle and Sign of Trigonometric Functions. JEE uses this to find exact values for angles beyond $90^{\circ}$ , e.g., $\sin 120^{\circ}$ .
Graphs of Trigonometric Functions. JEE may ask about number of solutions to equations like $\sin x = x$ using graphs.
Inverse Trigonometric Functions: Principal Values. JEE tests properties like $\sin^{- 1} x + \cos^{- 1} x = \frac{π}{2}$ .
Trigonometric Identities: Proving and Simplifying. JEE often asks to prove identities or simplify expressions in multiple-choice questions.
Trigonometric Ratios of Allied Angles. JEE uses these in problems involving multiple angles or simplification.
Maximum and Minimum Values of Trigonometric Expressions. JEE often asks for maximum value of $a \sin θ + b \cos θ + c$ .
Sine and Cosine Laws for Triangles. JEE tests these in geometry problems, often combined with area formulas.
Area of a Triangle: Trigonometric Formula. JEE often combines this with sine/cosine laws in triangle problems.
Periodicity and Symmetry of Trigonometric Functions. JEE tests these in function analysis and equation solving.
Solving Trigonometric Inequalities. JEE may ask for solution sets of inequalities like $\cos x \leq \frac{\sqrt{3}}{2}$ .
Trigonometric Substitutions in Integration. JEE tests these in calculus problems, often requiring simplification using trig identities.
Trigonometric Equations: Using Auxiliary Angle. JEE often uses this technique in trigonometric equations and optimization.
Heights and Distances: Angle of Elevation/Depression. JEE tests these in applied problems, often with two angles or multiple observations.
Trigonometric Series: Summation of Sine/Cosine Series. JEE may ask for sum of series like $\sin θ + \sin 2 θ + \dots + \sin n θ$ .

Common traps

Forgetting to convert degrees to radians when using calculus or arc length formulas. Angle Measurement: Degrees & Radians
Confusing opposite and adjacent sides for non-right triangles. Trigonometric Ratios: Definitions
Forgetting the plus sign in $1 + \tan^{2} θ = \sec^{2} θ$ . Pythagorean Identities
Mixing up signs: $\cos (A + B) = \cos A \cos B - \sin A \sin B$ (minus, not plus). Sum and Difference Formulas
Forgetting the multiple forms of $\cos 2 θ$ ; each is useful in different contexts. Double Angle Formulas
Forgetting the $\pm$ sign; the sign depends on the quadrant of $θ / 2$ . Half Angle Formulas
Forgetting the factor $\frac{1}{2}$ in product-to-sum formulas. Product-to-Sum and Sum-to-Product Formulas
Forgetting to consider all quadrants when solving $\sin θ = k$ . Solving Trigonometric Equations
Confusing $\sin 30^{\circ}$ with $\sin 60^{\circ}$ values. Trigonometric Ratios of Standard Angles
Forgetting that $\sec θ$ and $\csc θ$ have same sign as $\cos θ$ and $\sin θ$ respectively. Unit Circle and Sign of Trigonometric Functions
Forgetting that $\tan x$ has vertical asymptotes, not defined at those points. Graphs of Trigonometric Functions
Assuming $\sin^{- 1} (\sin x) = x$ for all $x$ ; it's only true in the principal range. Inverse Trigonometric Functions: Principal Values
Cross-multiplying without checking domain restrictions. Trigonometric Identities: Proving and Simplifying
Forgetting sign changes: $\cos (90^{\circ} + θ) = - \sin θ$ (negative). Trigonometric Ratios of Allied Angles

Topics in Trigonometry

Trigonometric Ratios and Identities
Compound and Multiple Angles
Trigonometric Equations
Inverse Trigonometric Functions
Properties of Triangles
Heights and Distances

Previous-year questions on Trigonometry

JEEGenie has 488 previous-year questions on Trigonometry. A few worked examples follow.

Year	Questions
2026	23
2025	21
2024	21
2023	28
2022	34
2021	37
2020	14
2019	68
2018	11
2017	9
2016	13
2015	8
2014	10
2013	12
2012	10
2011	3
2010	10
2009	9
2008	3
2007	6
2006	6
2005	5
2004	7
2003	6
2002	8
2000	4
1999	4
1998	4
1997	2
1996	5
1995	4
1994	3
1993	9
1992	1
1991	6
1990	3
1989	1
1988	6
1987	6
1986	7
1984	7
1983	7
1982	3
1981	6
1980	5
1979	6
1978	7

Question 1 medium

\sin (\frac{π}{18}) \sin (\frac{5 π}{18}) \sin (\frac{7 π}{18}) = K

, then the value of

\sin (\frac{10 K π}{3})

is:

$\frac{\sqrt{3} + 1}{2 \sqrt{2}}$
$\frac{\sqrt{3} - 1}{\sqrt{2}}$
$\frac{\sqrt{3}}{2}$
$\frac{1}{2}$

Answer: A

Explanation.

\sin \frac{π}{18} \sin \frac{5 π}{18} \sin \frac{7 π}{18} = \sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ}

Use the identity:

\sin θ \sin (60^{\circ} - θ) \sin (60^{\circ} + θ) = \frac{1}{4} \sin 3 θ

with

θ = 10^{\circ}

\sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ} = \frac{1}{4} \sin 30^{\circ} = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8}

So,

K = \frac{1}{8}

Now compute:

\sin (\frac{10 K π}{3}) = \sin (\frac{10 \cdot \frac{1}{8} π}{3}) = \sin (\frac{5 π}{12})

Using

\sin 75^{\circ} = \sin (45^{\circ} + 30^{\circ})

\sin 75^{\circ} = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} + 1}{2 \sqrt{2}}

Therefore,

\frac{\sqrt{3} + 1}{2 \sqrt{2}}

Answer: **A**

From NTA/IIT papers: JEE Main 2026 · 2 Apr Shift 1

Question 2 hard

If for

θ \in [- \frac{π}{3}, 0]

, the points

(x, y) = (3 \tan (θ + \frac{π}{3}), 2 \tan (θ + \frac{π}{6}))

lie on

x y + α x + β y + γ = 0

, then

α^{2} + β^{2} + γ^{2}

is equal to :

Answer: A

Explanation. Let

u = θ + \frac{π}{3}

, so

θ = u - \frac{π}{3}

. Then

θ + \frac{π}{6} = u - \frac{π}{6}

. Given

x = 3 \tan u

y = 2 \tan (u - \frac{π}{6})

. Using

\tan (A - B)

y = 2 \cdot \frac{\tan u - \frac{1}{\sqrt{3}}}{1 + \frac{\tan u}{\sqrt{3}}}

. Let

t = \tan u

. Then

x = 3 t

, and

y = 2 \cdot \frac{t - \frac{1}{\sqrt{3}}}{1 + \frac{t}{\sqrt{3}}} = 2 \cdot \frac{\sqrt{3} t - 1}{\sqrt{3} + t}

. Substitute

t = \frac{x}{3}

y = 2 \cdot \frac{\sqrt{3} \cdot \frac{x}{3} - 1}{\sqrt{3} + \frac{x}{3}} = 2 \cdot \frac{\frac{\sqrt{3} x}{3} - 1}{\frac{3 \sqrt{3} + x}{3}} = 2 \cdot \frac{\sqrt{3} x - 3}{3 \sqrt{3} + x}

. Cross-multiply:

y (3 \sqrt{3} + x) = 2 (\sqrt{3} x - 3) \Rightarrow x y + 3 \sqrt{3} y = 2 \sqrt{3} x - 6 \Rightarrow x y - 2 \sqrt{3} x + 3 \sqrt{3} y + 6 = 0

. Thus

α = - 2 \sqrt{3}

β = 3 \sqrt{3}

γ = 6

. Compute:

α^{2} + β^{2} + γ^{2} = 12 + 27 + 36 = 75

. The answer is option A.

From NTA/IIT papers: JEE Main 2025 · 7 Apr Shift 1

Question 3 easy

\sin x = - \frac{3}{5}

, where

π < x < \frac{3 π}{2}

, then

80 (\tan^{2} x - \cos x)

is equal to

Answer: A

Explanation. Since

π < x < \frac{3 π}{2}

x

is in the third quadrant, so

\sin x < 0

\cos x < 0

\tan x > 0

. Given

\sin x = - \frac{3}{5}

, we use

\sin^{2} x + \cos^{2} x = 1

\cos^{2} x = 1 - \frac{9}{25} = \frac{16}{25}

Since

\cos x < 0

\cos x = - \frac{4}{5}

Then

\tan x = \frac{\sin x}{\cos x} = \frac{- 3 / 5}{- 4 / 5} = \frac{3}{4}

\tan^{2} x = \frac{9}{16}

Now compute:

\tan^{2} x - \cos x = \frac{9}{16} - (- \frac{4}{5}) = \frac{9}{16} + \frac{4}{5}

= \frac{45 + 64}{80} = \frac{109}{80}

Therefore,

80 (\tan^{2} x - \cos x) = 80 \cdot \frac{109}{80} = 109

109

Answer: A) 109

From NTA/IIT papers: JEE Main 2024 · 8 Apr Shift 1

Question 4 medium

The value of

\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ}

is __________.

Answer: 4

Explanation. We need to find

\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ}

. Step 1: Use complementary angle identities. Note that

\tan 81^{\circ} = \tan (90^{\circ} - 9^{\circ}) = \cot 9^{\circ}

and

\tan 63^{\circ} = \tan (90^{\circ} - 27^{\circ}) = \cot 27^{\circ}

. So the expression becomes:

\tan 9^{\circ} + \cot 9^{\circ} - (\tan 27^{\circ} + \cot 27^{\circ})

Step 2: Simplify

\tan θ + \cot θ

\tan θ + \cot θ = \frac{\sin θ}{\cos θ} + \frac{\cos θ}{\sin θ} = \frac{\sin^{2} θ + \cos^{2} θ}{\sin θ \cos θ} = \frac{1}{\sin θ \cos θ} = \frac{2}{\sin 2 θ}

Step 3: Apply this formula.

(\tan 9^{\circ} + \cot 9^{\circ}) - (\tan 27^{\circ} + \cot 27^{\circ}) = \frac{2}{\sin 18^{\circ}} - \frac{2}{\sin 54^{\circ}}

Step 4: Use known values.

\sin 18^{\circ} = \frac{\sqrt{5} - 1}{4}

and

\sin 54^{\circ} = \cos 36^{\circ} = \frac{\sqrt{5} + 1}{4}

. So:

\frac{2}{\frac{\sqrt{5} - 1}{4}} - \frac{2}{\frac{\sqrt{5} + 1}{4}} = \frac{8}{\sqrt{5} - 1} - \frac{8}{\sqrt{5} + 1}

Step 5: Combine the fractions.

= 8 \cdot \frac{(\sqrt{5} + 1) - (\sqrt{5} - 1)}{(\sqrt{5} - 1) (\sqrt{5} + 1)} = 8 \cdot \frac{2}{5 - 1} = 8 \cdot \frac{2}{4} = 8 \cdot \frac{1}{2} = 4

Therefore,

\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ} = 4

From NTA/IIT papers: JEE Main 2023 · 6 Apr Shift 2

Question 5 hard

Let

α

and

β

be real numbers such that

- \frac{π}{4} < β < 0 < α < \frac{π}{4}

. If

\sin (α + β) = \frac{1}{3}

and

\cos (α - β) = \frac{2}{3}

, then the greatest integer less than or equal to

{(\frac{\sin α}{\cos β} + \frac{\cos β}{\sin α} + \frac{\cos α}{\sin β} + \frac{\sin β}{\cos α})}^{2}

is _____.

Answer: 1

Explanation. Let

S = \frac{\sin α}{\cos β} + \frac{\cos β}{\sin α} + \frac{\cos α}{\sin β} + \frac{\sin β}{\cos α} .

Group terms:

S = (\frac{\sin α}{\cos β} + \frac{\sin β}{\cos α}) + (\frac{\cos β}{\sin α} + \frac{\cos α}{\sin β}) .

Now

\frac{\sin α}{\cos β} + \frac{\sin β}{\cos α} = \frac{\sin α \cos α + \sin β \cos β}{\cos α \cos β} = \frac{\frac{1}{2} (\sin 2 α + \sin 2 β)}{\cos α \cos β} .

Using

\sin 2 α + \sin 2 β = 2 \sin (α + β) \cos (α - β)

\frac{\sin α}{\cos β} + \frac{\sin β}{\cos α} = \frac{\sin (α + β) \cos (α - β)}{\cos α \cos β} .

Similarly,

\frac{\cos β}{\sin α} + \frac{\cos α}{\sin β} = \frac{\cos β \sin β + \cos α \sin α}{\sin α \sin β} = \frac{\sin (α + β) \cos (α - β)}{\sin α \sin β} .

Thus

S = \sin (α + β) \cos (α - β) (\frac{1}{\cos α \cos β} + \frac{1}{\sin α \sin β}) .

But

\frac{1}{\cos α \cos β} + \frac{1}{\sin α \sin β} = \frac{\sin α \sin β + \cos α \cos β}{\sin α \sin β \cos α \cos β} = \frac{\cos (α - β)}{\sin α \sin β \cos α \cos β} .

S = \frac{\sin (α + β) \cos^{2} (α - β)}{\sin α \sin β \cos α \cos β} .

Given

\sin (α + β) = \frac{1}{3}

\cos (α - β) = \frac{2}{3}

S = \frac{\frac{1}{3} \cdot \frac{4}{9}}{\sin α \sin β \cos α \cos β} = \frac{4}{27 \sin α \sin β \cos α \cos β} .

Now

\sin α \sin β \cos α \cos β = \frac{1}{4} (\sin 2 α) (\sin 2 β) .

Also

\sin 2 α \sin 2 β = \cos^{2} (α - β) - \cos^{2} (α + β) .

Since

\cos (α - β) = \frac{2}{3}

\cos^{2} (α - β) = \frac{4}{9}

. And

\sin (α + β) = \frac{1}{3}

, so

\cos^{2} (α + β) = 1 - \frac{1}{9} = \frac{8}{9}

. Thus

\sin 2 α \sin 2 β = \frac{4}{9} - \frac{8}{9} = - \frac{4}{9} .

Hence

\sin α \sin β \cos α \cos β = \frac{1}{4} (- \frac{4}{9}) = - \frac{1}{9} .

Therefore

S = \frac{4}{27 (- \frac{1}{9})} = \frac{4}{- 3} = - \frac{4}{3} .

S^{2} = \frac{16}{9} .

The greatest integer less than or equal to

\frac{16}{9}

1

. Answer: 1

From NTA/IIT papers: JEE Advanced 2022

Frequently asked questions

How much of JEE Main comes from Trigonometry?: JEEGenie estimates about 5% of JEE Main marks come from Trigonometry, based on recent papers. The share varies from year to year.
Which formulas matter most in Trigonometry?: The Trigonometry sheet on this page has 23 formula entries, starting with Angle Measurement: Degrees & Radians, Trigonometric Ratios: Definitions, Pythagorean Identities, Sum and Difference Formulas, Double Angle Formulas. Each one comes with when to use it and the trap to avoid.
What topics does Trigonometry cover for JEE?: Trigonometry covers Trigonometric Ratios and Identities, Compound and Multiple Angles, Trigonometric Equations, Inverse Trigonometric Functions, Properties of Triangles, Heights and Distances.
How should I revise Trigonometry before the exam?: Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Trigonometry questions on JEEGenie.
How many previous-year questions are there on Trigonometry?: JEEGenie has 488 previous-year questions on Trigonometry, and this page shows a few with worked explanations.
Is practice on Trigonometry free on JEEGenie?: Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.