Trigonometric Ratios and Identities

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JEE Main 2014

Let $f_{k} (x) = \frac{1}{k} (sin^{k} x + cos^{k} x)$ where $x \in R$ and $k \geq 1$ . Then $f_{4} (x) - f_{6} (x)$ equals

1/4

(B)

1/12

(C)

1/6

(D)

1/3

JEE Main 2013

LEVELJEE Main

The expression $\frac{t a n A}{1 - c o t A} + \frac{c o t A}{1 - t a n A}$ can be written as

(A)

sin A cos A + 1

(B)

sec A csc A + 1

(C)

tan A + cot A

(D)

sec A + csc A

JEE Main 2012

LEVELJEE Main

In a $△ P QR$ , if $3 sin P + 4 cos Q = 6$ and $4 sin Q + 3 cos P = 1$ , then the angle $R$ is equal to

(A)

5 π /6

(B)

π /6

(C)

π /4

(D)

3 π /4

JEE Main 2011

LEVELJEE Main

If $A = sin^{2} x + cos^{4} x$ , then for all real $x$ :

(A)

13/16 \leq A \leq 1

(B)

1 \leq A \leq 2

(C)

3/4 \leq A \leq 13/16

(D)

3/4 \leq A \leq 1

JEE Main 2009

LEVELJEE Main

If $cos (β - γ) + cos (γ - α) + cos (α - β) = - 3/2$ , then

(A)

A is false and B is true

(B)

both A and B are true

(C)

both A and B are false

(D)

A is true and B is false

JEE Main 2007

LEVELJEE Main

If $p$ and $q$ are positive real numbers such that $p^{2} + q^{2} = 1$ , then the maximum value of $(p + q)$ is

2

(D)

JEE Main 2006

LEVELJEE Main

If $0 < x < π$ and $cos x + sin x = 1/2$ , then $tan x$ is

(A)

\frac{( 1 - 7 )}{4}

(B)

\frac{( 4 - 7 )}{3}

(C)

- \frac{( 4 + 7 )}{3}

(D)

\frac{( 1 + 7 )}{4}

JEE Main 2004

LEVELJEE Main

Let $α, β$ be such that $π < α - β < 3 π$ . If $sin α + sin β = - 21/65$ and $cos α + cos β = - 27/65$ , then the value of $cos \frac{α - β}{2}$ is

(A)

- 6/65

(B)

3/ 130

(C)

6/65

(D)

- 3/ 130

JEE Main 2004

LEVELJEE Main

If $u = a^{2} cos^{2} θ + b^{2} sin^{2} θ + a^{2} sin^{2} θ + b^{2} cos^{2} θ$ then the difference between the maximum and minimum values of $u^{2}$ is given by

(A)

(a - b)^{2}

(B)

2 a^{2} + b^{2}

(C)

(a + b)^{2}

(D)

2 (a^{2} + b^{2})

JEE Main 2004

LEVELJEE Main

If $t_{1}$ and $t_{2}$ are the times of flight of two particles having the same initial velocity u and range R on the horizontal, then $t_{1}^{2} + t_{2}^{2}$ is equal to

(A)

(B)

4 u^{2} / g^{2}

(C)

u^{2} /2 g

(D)

u^{2} / g

JEE Main 2002

LEVELBoard

The period of $sin^{2} θ$ is

(A)

π^{2}

(B)

π

(C)

2 π

(D)

π /2

JEE Main 2002

LEVELJEE Main

A bead of weight w can slide on smooth circular wire in a vertical plane. The bead is attached by a light thread to the highest point of the wire and in equilibrium, the thread is taut and make an angle $θ$ with the vertical then tension of the thread and reaction of the wire on the bead are

(A)

T = w cos θ, R = w tan θ

(B)

T = 2 w cos θ, R = w

(C)

T = w, R = w sin θ

(D)

T = w sin θ, R = w cot θ

JEE Advanced 2016

LEVELJEE Main

The value of $\sum_{k = 1}^{13} \frac{1}{s i n ( \frac{π}{4} + \frac{( k - 1 ) π}{6} ) s i n ( \frac{π}{4} + \frac{k π}{6} )}$ is equal to

(A)

3 - 3

(B)

2 (3 - 3)

(C)

2 (3 - 1)

(D)

2 (2 - 3)

JEE Advanced 2009

LEVELJEE Main

If $\frac{s i n ^{4} x}{2} + \frac{c o s ^{4} x}{3} = \frac{1}{5}$ , then

* Multiple Correct Options

(A)

tan^{2} x = 2/3

(B)

\frac{s i n ^{8} x}{8} + \frac{c o s ^{8} x}{27} = \frac{1}{125}

(C)

tan^{2} x = 1/3

(D)

\frac{s i n ^{8} x}{8} + \frac{c o s ^{8} x}{27} = \frac{2}{125}

JEE Advanced 2009

LEVELJEE Main

For $0 < θ < π /2$ , the solution(s) of $\sum_{m = 1}^{6} csc (θ + \frac{( m - 1 ) π}{4}) csc (θ + \frac{mπ}{4}) = 42$ is (are)

* Multiple Correct Options

(A)

π /4

(B)

π /6

(C)

π /12

(D)

5 π /12

JEE Advanced 2006

LEVELJEE Main

Let $θ \in (0, π /4)$ and $t_{1} = (tan θ)^{t a n θ}, t_{2} = (tan θ)^{c o t θ}, t_{3} = (cot θ)^{t a n θ}$ and $t_{4} = (cot θ)^{c o t θ}$ , then

(A)

t_{1} > t_{2} > t_{3} > t_{4}

(B)

t_{4} > t_{3} > t_{1} > t_{2}

(C)

t_{3} > t_{1} > t_{2} > t_{4}

(D)

t_{2} > t_{3} > t_{1} > t_{4}

JEE Advanced 2004

LEVELJEE Main

Given both $θ$ and $ϕ$ are acute angles and $sin θ = 1/2, cos ϕ = 1/3$ , then the value of $θ + ϕ$ belongs to

(A)

(π /3, π /2]

(B)

(π /2, 2 π /3]

(C)

(2 π /3, 5 π /6]

(D)

(5 π /6, π]

JEE Advanced 2002

LEVELJEE Main

The number of integral values of $k$ for which the equation $7 cos x + 5 sin x = 2 k + 1$ has a solution is

(A)

4

(B)

8

(C)

10

(D)

12

JEE Advanced 2001

LEVELJEE Main

The maximum value of $(cos α_{1}) \cdot (cos α_{2}) \dots (cos α_{n})$ , under the restrictions $0 \leq α_{1}, α_{2}, \dots, α_{n} \leq π /2$ and $(cot α_{1}) \cdot (cot α_{2}) \dots (cot α_{n}) = 1$ is

(A)

1/ 2^{n /2}

(B)

1/ 2^{n}

(C)

1/2 n

(D)

1

JEE Advanced 2001

LEVELJEE Main

If $α + β = π /2$ and $β + γ = α$ , then $tan α$ equals

(A)

2 (tan β + tan γ)

(B)

tan β + tan γ

(C)

tan β + 2 tan γ

(D)

2 tan β + tan γ

JEE Advanced 2000

LEVELBoard

Let $f (θ) = sin θ (sin θ + sin 3 θ)$ . Then $f (θ)$ is

(A)

\geq 0

only when

θ \geq 0

(B)

\leq 0

for all real

θ

(C)

\geq 0

for all real

θ

(D)

\leq 0

only when

θ \leq 0

JEE Advanced 1999

LEVELJEE Main

For a positive integer $n$ , let $f_{n} (θ) = (tan \frac{θ}{2}) (1 + sec θ) (1 + sec 2 θ) (1 + sec 4 θ) \dots (1 + sec 2^{n} θ)$ . Then

* Multiple Correct Options

(A)

f_{2} (π /16) = 1

(B)

f_{3} (π /32) = 1

(C)

f_{4} (π /64) = 1

(D)

f_{5} (π /128) = 1

JEE Advanced 1998

LEVELJEE Main

Which of the following number(s) is/are rational?

(A)

sin 1 5^{\circ}

(B)

cos 1 5^{\circ}

(C)

sin 1 5^{\circ} cos 1 5^{\circ}

(D)

sin 1 5^{\circ} cos 7 5^{\circ}

JEE Advanced 1996

LEVELBoard

$sec^{2} θ = \frac{4 x y}{( x + y ) ^{2}}$ is true if and only if

(A)

x + y \neq = 0

(B)

x = y, x \neq = 0

(C)

x = y

(D)

x \neq = 0, y \neq = 0

JEE Advanced 1995

LEVELJEE Main

$3 (sin x - cos x)^{4} + 6 (sin x + cos x)^{2} + 4 (sin^{6} x + cos^{6} x) =$

(A)

11

(B)

12

(C)

13

(D)

14

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The expression 1−cotAtanA​+1−tanAcotA​ can be written as

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a △PQR, if 3sinP+4cosQ=6 and 4sinQ+3cosP=1, then the angle R is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=sin2x+cos4x, then for all real x:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If cos(β−γ)+cos(γ−α)+cos(α−β)=−3/2, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If p and q are positive real numbers such that p2+q2=1, then the maximum value of (p+q) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 0<x<π and cosx+sinx=1/2, then tanx is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β be such that π<α−β<3π. If sinα+sinβ=−21/65 and cosα+cosβ=−27/65, then the value of cos2α−β​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If u=a2cos2θ+b2sin2θ​+a2sin2θ+b2cos2θ​ then the difference between the maximum and minimum values of u2 is given by

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If t1​ and t2​ are the times of flight of two particles having the same initial velocity u and range R on the horizontal, then t12​+t22​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The period of sin2θ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of ∑k=113​sin(4π​+6(k−1)π​)sin(4π​+6kπ​)1​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 2sin4x​+3cos4x​=51​, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For 0<θ<π/2, the solution(s) of ∑m=16​csc(θ+4(m−1)π​)csc(θ+4mπ​)=42​ is (are)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let θ∈(0,π/4) and t1​=(tanθ)tanθ,t2​=(tanθ)cotθ,t3​=(cotθ)tanθ and t4​=(cotθ)cotθ, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given both θ and ϕ are acute angles and sinθ=1/2,cosϕ=1/3, then the value of θ+ϕ belongs to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of integral values of k for which the equation 7cosx+5sinx=2k+1 has a solution is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The maximum value of (cosα1​)⋅(cosα2​)…(cosαn​), under the restrictions 0≤α1​,α2​,…,αn​≤π/2 and (cotα1​)⋅(cotα2​)…(cotαn​)=1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α+β=π/2 and β+γ=α, then tanα equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(θ)=sinθ(sinθ+sin3θ). Then f(θ) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For a positive integer n, let fn​(θ)=(tan2θ​)(1+secθ)(1+sec2θ)(1+sec4θ)…(1+sec2nθ). Then

Which of the following number(s) is/are rational?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } sec2θ=(x+y)24xy​ is true if and only if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 3(sinx−cosx)4+6(sinx+cosx)2+4(sin6x+cos6x)=

Let $f_{k} (x) = \frac{1}{k} (sin^{k} x + cos^{k} x)$ where $x \in R$ and $k \geq 1$ . Then $f_{4} (x) - f_{6} (x)$ equals

The expression $\frac{t a n A}{1 - c o t A} + \frac{c o t A}{1 - t a n A}$ can be written as

In a $△ P QR$ , if $3 sin P + 4 cos Q = 6$ and $4 sin Q + 3 cos P = 1$ , then the angle $R$ is equal to

If $A = sin^{2} x + cos^{4} x$ , then for all real $x$ :

If $cos (β - γ) + cos (γ - α) + cos (α - β) = - 3/2$ , then

If $p$ and $q$ are positive real numbers such that $p^{2} + q^{2} = 1$ , then the maximum value of $(p + q)$ is

If $0 < x < π$ and $cos x + sin x = 1/2$ , then $tan x$ is

Let $α, β$ be such that $π < α - β < 3 π$ . If $sin α + sin β = - 21/65$ and $cos α + cos β = - 27/65$ , then the value of $cos \frac{α - β}{2}$ is

If $u = a^{2} cos^{2} θ + b^{2} sin^{2} θ + a^{2} sin^{2} θ + b^{2} cos^{2} θ$ then the difference between the maximum and minimum values of $u^{2}$ is given by

If $t_{1}$ and $t_{2}$ are the times of flight of two particles having the same initial velocity u and range R on the horizontal, then $t_{1}^{2} + t_{2}^{2}$ is equal to

The period of $sin^{2} θ$ is

The value of $\sum_{k = 1}^{13} \frac{1}{s i n ( \frac{π}{4} + \frac{( k - 1 ) π}{6} ) s i n ( \frac{π}{4} + \frac{k π}{6} )}$ is equal to

If $\frac{s i n ^{4} x}{2} + \frac{c o s ^{4} x}{3} = \frac{1}{5}$ , then

For $0 < θ < π /2$ , the solution(s) of $\sum_{m = 1}^{6} csc (θ + \frac{( m - 1 ) π}{4}) csc (θ + \frac{mπ}{4}) = 42$ is (are)

Let $θ \in (0, π /4)$ and $t_{1} = (tan θ)^{t a n θ}, t_{2} = (tan θ)^{c o t θ}, t_{3} = (cot θ)^{t a n θ}$ and $t_{4} = (cot θ)^{c o t θ}$ , then

Given both $θ$ and $ϕ$ are acute angles and $sin θ = 1/2, cos ϕ = 1/3$ , then the value of $θ + ϕ$ belongs to

The number of integral values of $k$ for which the equation $7 cos x + 5 sin x = 2 k + 1$ has a solution is

The maximum value of $(cos α_{1}) \cdot (cos α_{2}) \dots (cos α_{n})$ , under the restrictions $0 \leq α_{1}, α_{2}, \dots, α_{n} \leq π /2$ and $(cot α_{1}) \cdot (cot α_{2}) \dots (cot α_{n}) = 1$ is

If $α + β = π /2$ and $β + γ = α$ , then $tan α$ equals

Let $f (θ) = sin θ (sin θ + sin 3 θ)$ . Then $f (θ)$ is

For a positive integer $n$ , let $f_{n} (θ) = (tan \frac{θ}{2}) (1 + sec θ) (1 + sec 2 θ) (1 + sec 4 θ) \dots (1 + sec 2^{n} θ)$ . Then

$sec^{2} θ = \frac{4 x y}{( x + y ) ^{2}}$ is true if and only if

$3 (sin x - cos x)^{4} + 6 (sin x + cos x)^{2} + 4 (sin^{6} x + cos^{6} x) =$