Core Syllabus Module

Trigonometry

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Chapter Elite-20 Challenge Zone

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JEE Advanced 2019

LEVELJEE Advanced

In a non-right-angled triangle $P QR$ , let $p, q, r$ denote the lengths of the sides opposite to the angles at $P, Q, R$ respectively. The median from $R$ meets the side $P Q$ at $S$ , the perpendicular from $P$ meets the side $QR$ at $E$ , and $R S$ and $P E$ intersect at $O$ . If $p = 3, q = 1$ , and the radius of the circumcircle of the $Δ P QR$ equals 1, then which of the following options is/are correct?

* Multiple Correct Options

(A)

Area of

Δ S O E = \frac{3}{12}

(B)

Radius of incircle of

Δ P QR = \frac{3}{2} (2 - 3)

(C)

Length of

R S = \frac{7}{2}

(D)

Length of

O E = \frac{1}{6}

JEE Advanced 1999

LEVELJEE Advanced

Let $A B C$ be a triangle having $O$ and $I$ as its circumcenter and in centre respectively. If $R$ and $r$ are the circumradius and the inradius, respectively, then prove that $(I O)^{2} = R^{2} - 2 R r$ . Further show that the triangle $B I O$ is a right-angled triangle if and only if $b$ is arithmetic mean of $a$ and $c$ .

JEE Advanced 1979

LEVELJEE Advanced

(a) A balloon is observed simultaneously from three points $A, B$ and $C$ on a straight road directly beneath it. The angular elevation at $B$ is twice that at $A$ and the angular elevation at $C$ is thrice that at $A$ . If the distance between $A$ and $B$ is $a$ and the distance between $B$ and $C$ is $b$ , find the height of the balloon in terms of $a$ and $b$ . (b) Find the area of the smaller part of a disc of radius $10$ cm, cut off by a chord $A B$ which subtends an angle of $22 \frac{1}{2}^{\circ}$ at the circumference.

JEE Advanced 2000

LEVELJEE Advanced

Let $A B C$ be a triangle with incentre $I$ and inradius $r$ . Let $D, E, F$ be the feet of the perpendiculars from $I$ to the sides $B C, C A$ and $A B$ respectively. If $r_{1}, r_{2}$ and $r_{3}$ are the radii of circles inscribed in the quadrilaterals $A F I E, B D I F$ and $C E I D$ respectively, prove that $\frac{r _{1}}{r - r _{1}} + \frac{r _{2}}{r - r _{2}} + \frac{r _{3}}{r - r _{3}} = \frac{r _{1} r _{2} r _{3}}{( r - r _{1} ) ( r - r _{2} ) ( r - r _{3} )}$ .

JEE Advanced 2009

LEVELJEE Advanced

In a triangle $A B C$ with fixed base $B C$ , the vertex $A$ moves such that $cos B + cos C = 4 sin^{2} \frac{A}{2}$ . If $a, b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B$ and $C$ , respectively, then

* Multiple Correct Options

(A)

b + c = 4 a

(B)

b + c = 2 a

(C)

locus of point

A

is an ellipse

(D)

locus of point

A

is a pair of straight lines

JEE(ADVANCED)-201

LEVELJEE Advanced

Let $a, b, c$ be three non-zero real numbers such that the equation $3 a cos x + 2 b sin x = c, x \in (- \frac{π}{2}, \frac{π}{2})$ has two distinct real roots $α$ and $β$ with $α + β = \frac{π}{3}$ . Then, the value of $\frac{b}{a}$ is ________.

JEE Main 2023 (10 Apr Shift 2)

LEVELJEE Advanced

In the figure, $θ_{1} + θ_{2} = \frac{π}{2}$ and $3 (B E) = 4 (A B)$ . If the area of $Δ C A B$ is $23 - 3 unit^{2}$ , when $\frac{θ _{2}}{θ _{1}}$ is the largest, then the perimeter (in unit) of $Δ C E D$ is equal to

JEE Advanced 1978

LEVELJEE Advanced

A triangle $A B C$ has sides $A B = A C = 5$ cm and $B C = 6$ cm. Triangle $A^{'} B^{'} C^{'}$ is the reflection of the triangle $A B C$ in a line parallel to $A B$ placed at a distance $2$ cm from $A B$ , outside the triangle $A B C$ . Triangle $A^{''} B^{''} C^{''}$ is the reflection of the triangle $A^{'} B^{'} C^{'}$ in a line parallel to $B^{'} C^{'}$ placed at a distance $2$ cm from $B^{'} C^{'}$ outside the triangle $A^{'} B^{'} C^{'}$ . Find the distance between $A$ and $A^{''}$ .

JEE Advanced 2020

LEVELJEE Advanced

Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$ , respectively. If $tan \frac{X}{2} + tan \frac{Z}{2} = \frac{2 y}{x + y + z}$ , then which of the following statements is/are TRUE?

* Multiple Correct Options

(A)

2 Y = X + Z

(B)

Y = X + Z

(C)

tan \frac{X}{2} = \frac{x}{y + z}

(D)

x^{2} + z^{2} - y^{2} = x z

JEE Main 2025 April

LEVELJEE Advanced

The number of solutions of the equation $2 x + 3 tan x = π, x \in [- 2 π, 2 π] - {\pm \frac{π}{2}, \pm \frac{3 π}{2}}$ is

JEE Main 2024 (27 Jan Shift 1)

LEVELJEE Advanced

Let the set of all $a \in R$ such that the equation $cos 2 x + a sin x = 2 a - 7$ has a solution be $[p, q]$ and $r = tan 9^{\circ} - tan 2 7^{\circ} - \frac{1}{c o t 6 3 ^{\circ}} + tan 8 1^{\circ}$ , then $pq r$ is equal to

JEE Advanced 1998

LEVELJEE Advanced

A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose $6 0^{\circ}$ and $3 0^{\circ}$ are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is at the points $P$ and $Q$ respectively on its path. Let $θ$ be the angle of elevation of the bird when it is a point on the arc of the circle exactly midway between $P$ and $Q$ . Find the numerical value of $tan^{2} θ$ . (Assume that the observer is not inside the vertical projection of the path of the bird.)

JEE Advanced 1990

LEVELJEE Advanced

A vertical tower $P Q$ stands at a point $P$ . Points $A$ and $B$ are located to the South and East of $P$ respectively. $M$ is the mid point of $A B$ . $P A M$ is an equilateral triangle; and $N$ is the foot of the perpendicular from $P$ on $A B$ . Let $A N = 20$ metres and the angle of elevation of the top of the tower at $N$ is $tan^{- 1} (2)$ . Determine the height of the tower and the angles of elevation of the top of the tower at $A$ and $B$ .

JEE Advanced 1985

LEVELJEE Advanced

In a triangle $A B C$ , the median to the side $B C$ is of length $\frac{1}{11 - 6 3}$ and it divides the angle $A$ into angles $3 0^{\circ}$ and $4 5^{\circ}$ . Find the length of the side $B C$ .

JEE Main 2026 (22 January Shift 2)

LEVELJEE Advanced

Let $cos (α + β) = - \frac{1}{10}$ and $sin (α - β) = \frac{3}{8}$ , where $0 < α < \frac{π}{3}$ and $0 < β < \frac{π}{4}$ . If $tan 2 α = \frac{3 ( 1 - r 5 )}{11 ( s + 5 )}, r, s \in N$ , then $r + s$ is equal to ......... .

JEE Advanced 2023

LEVELJEE Advanced

Comprehension Passage

Consider on obtuse angled triangle

A B C

in which the difference between the largest and the smallest angle is

\frac{π}{2}

and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.

Question 1:

Let $a$ be the area of the triangle $A B C$ . Then the value of $(64 a)^{2}$ is

Question 2:

Then the inradius of the triangle $A B C$ is

JEE Main 2025 (January)

LEVELJEE Advanced

Let $A = {x \in (0, π) - {\frac{π}{2}} : lo g_{(2/ π)} ∣ sin x ∣ + lo g_{(2/ π)} ∣ cos x ∣ = 2}$ and $B = {x \geq 0 : x (x - 4) - 3∣ x - 2∣ + 6 = 0}$ . Then $n (A \cup B)$ is equal to :

A circle passes through three points $A, B$ and $C$ with the line segment $A C$ as its diameter. A line passing through $A$ intersects the chord $B C$ at a point $D$ inside the circle. If angles $D A B$ and $C A B$ are $α$ and $β$ respectively and the distance between the point $A$ and the mid point of the line segment $D C$ is $d$ , prove that the area of the circle is $\frac{π d ^{2} c o s ^{2} α}{c o s ^{2} α + c o s ^{2} β + 2 c o s α c o s β c o s ( β - α )}$ .

JEE Main 2025 April

LEVELJEE Advanced

The number of solutions of the equation $cos 2 θ cos \frac{θ}{2} + cos \frac{5 θ}{2} = 2 cos^{3} \frac{5 θ}{2}$ in $[- \frac{π}{2}, \frac{π}{2}]$ is :

(i) $P Q$ is a vertical tower. $P$ is the foot and $Q$ is the top of the tower. $A, B, C$ are three points in the horizontal plane through $P$ . The angles of elevation of $Q$ from $A, B, C$ are equal, and each is equal to $θ$ . The sides of the triangle $A B C$ are $a, b, c$ ; and the area of the triangle $A B C$ is $Δ$ . Show that the height of the tower is $\frac{ab c t a n θ}{4Δ}$ . (ii) $A B$ is a vertical pole. The end $A$ is on the level ground. $C$ is the middle point of $A B$ . $P$ is a point on the level ground. The portion $C B$ subtends an angle $β$ at $P$ . If $A P = n A B$ , then show that $tan β = \frac{n}{2 n ^{2} + 1}$ .

Micro-Topic Strategy Matrix

Trigonometric Ratios and Identities

Trigonometric Functions of Compound Angles

Multiple and Sub-multiple Angles

Conditional Identities

General Solution of Trigonometric Equations

Properties of Triangles

Heights and Distances

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solutions of the equation 2x+3tanx=π,x∈[−2π,2π]−{±2π​,±23π​} is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the set of all a∈R such that the equation cos2x+asinx=2a−7 has a solution be [p,q] and r=tan9∘−tan27∘−cot63∘1​+tan81∘, then pqr is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle ABC, the median to the side BC is of length 11−63​​1​ and it divides the angle A into angles 30∘ and 45∘. Find the length of the side BC.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let cos(α+β)=−101​ and sin(α−β)=83​, where 0<α<3π​ and 0<β<4π​. If tan2α=11​(s+5​)3(1−r5​)​,r,s∈N, then r+s is equal to ......... .

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a be the area of the triangle ABC. Then the value of (64a)2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Then the inradius of the triangle ABC is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A={x∈(0,π)−{2π​}:log(2/π)​∣sinx∣+log(2/π)​∣cosx∣=2} and B={x≥0:x​(x​−4)−3∣x​−2∣+6=0}. Then n(A∪B) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solutions of the equation cos2θcos2θ​+cos25θ​=2cos325θ​ in [−2π​,2π​] is :

The number of solutions of the equation $2 x + 3 tan x = π, x \in [- 2 π, 2 π] - {\pm \frac{π}{2}, \pm \frac{3 π}{2}}$ is

Let the set of all $a \in R$ such that the equation $cos 2 x + a sin x = 2 a - 7$ has a solution be $[p, q]$ and $r = tan 9^{\circ} - tan 2 7^{\circ} - \frac{1}{c o t 6 3 ^{\circ}} + tan 8 1^{\circ}$ , then $pq r$ is equal to

In a triangle $A B C$ , the median to the side $B C$ is of length $\frac{1}{11 - 6 3}$ and it divides the angle $A$ into angles $3 0^{\circ}$ and $4 5^{\circ}$ . Find the length of the side $B C$ .

Let $cos (α + β) = - \frac{1}{10}$ and $sin (α - β) = \frac{3}{8}$ , where $0 < α < \frac{π}{3}$ and $0 < β < \frac{π}{4}$ . If $tan 2 α = \frac{3 ( 1 - r 5 )}{11 ( s + 5 )}, r, s \in N$ , then $r + s$ is equal to ......... .

Let $a$ be the area of the triangle $A B C$ . Then the value of $(64 a)^{2}$ is

Then the inradius of the triangle $A B C$ is

Let $A = {x \in (0, π) - {\frac{π}{2}} : lo g_{(2/ π)} ∣ sin x ∣ + lo g_{(2/ π)} ∣ cos x ∣ = 2}$ and $B = {x \geq 0 : x (x - 4) - 3∣ x - 2∣ + 6 = 0}$ . Then $n (A \cup B)$ is equal to :

The number of solutions of the equation $cos 2 θ cos \frac{θ}{2} + cos \frac{5 θ}{2} = 2 cos^{3} \frac{5 θ}{2}$ in $[- \frac{π}{2}, \frac{π}{2}]$ is :