General Solution of Trigonometric Equations

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JEE Main 2016

If $0 \leq x < 2 π$ , then the number of real values of $x$ , which satisfy the equation $cos x + cos 2 x + cos 3 x + cos 4 x = 0$ is

7

(B)

9

(C)

3

(D)

5

JEE Main 2006

LEVELBoard

The number of values of $x$ in the interval $[0, 3 π]$ satisfying the equation $2 sin^{2} x + 5 sin x - 3 = 0$ is

(A)

4

(B)

6

(C)

1

(D)

2

JEE Main 2002

LEVELJEE Main

The number of solution of $tan x + sec x = 2 cos x$ in $[0, 2 π)$ is

(A)

2

(B)

3

(C)

0

(D)

1

JEE Advanced 2016

LEVELJEE Main

Let $S = {x \in (- π, π) : x \neq = 0, \pm π /2}$ . The sum of all distinct solutions of the equation $3 sec x + csc x + 2 (tan x - cot x) = 0$ in the set $S$ is equal to

(A)

7 π /9

(B)

- 2 π /9

(C)

0

(D)

5 π /9

JEE Advanced 2014

LEVELJEE Main

For $x \in (0, π)$ , the equation $sin x + 2 sin 2 x - sin 3 x = 3$ has

(A)

infinitely many solutions

Let $θ, ϕ \in [0, 2 π]$ be such that $2 cos θ (1 - sin ϕ) = sin^{2} θ (tan \frac{θ}{2} + cot \frac{θ}{2}) cos ϕ - 1$ , $tan (2 π - θ) > 0$ and $- 1 < sin θ < - 3 /2$ , then $ϕ$ cannot satisfy

* Multiple Correct Options

(A)

0 < ϕ < π /2

(B)

π /2 < ϕ < 4 π /3

(C)

4 π /3 < ϕ < 3 π /2

(D)

3 π /2 < ϕ < 2 π

JEE Advanced 2011

LEVELJEE Main

Let $P = {θ : sin θ - cos θ = 2 cos θ}$ and $Q = {θ : sin θ + cos θ = 2 sin θ}$ be two sets. Then

(A)

P \subset Q

and

Q - P \neq = ϕ

(B)

Q \neq \subset P

(C)

P \neq \subset Q

(D)

P = Q

JEE Advanced 2007

LEVELJEE Main

The number of solutions of the pair of equations $2 sin^{2} θ - cos 2 θ = 0$ , $2 cos^{2} θ - 3 sin θ = 0$ in the interval $[0, 2 π]$ is

The values of $θ \in (0, 2 π)$ for which $2 sin^{2} θ - 5 sin θ + 2 > 0$ , are

(A)

(0, π /6) \cup (5 π /6, 2 π)

(B)

(π /8, 5 π /6)

(C)

(0, π /8) \cup (5 π /6, 2 π)

(D)

(41 π /48, π)

JEE Advanced 2005

LEVELJEE Main

$cos (α - β) = 1$ and $cos (α + β) = 1/ e$ where $α, β \in [- π, π]$ . Pairs of $α, β$ which satisfy both the equations is/are

(A)

0

(B)

1

(C)

2

(D)

4

JEE Advanced 1998

LEVELJEE Main

The number of values of $x$ in the interval $[0, 5 π]$ satisfying the equation $3 sin^{2} x - 7 sin x + 2 = 0$ is

(A)

0

(B)

5

(C)

6

(D)

10

JEE Advanced 1998

LEVELJEE Main

The number of values of $x$ where the function $f (x) = cos x + cos (2 x)$ attains its maximum is

The general values of $θ$ satisfying the equation $2 sin^{2} θ - sin θ - 2 = 0$ is

(A)

nπ + (- 1)^{n} π /6

(B)

nπ + (- 1)^{n} π /2

(C)

nπ + (- 1)^{n} 5 π /6

(D)

nπ + (- 1)^{n} 7 π /6

JEE Advanced 1993

LEVELJEE Main

Number of solutions of the equation $tan x + sec x = 2 cos x$ lying in the interval $[0, 2 π]$ is :

(A)

0

(B)

1

(C)

2

(D)

3

JEE Advanced 1989

LEVELJEE Main

The general solution of $sin x - 3 sin 2 x + sin 3 x = cos x - 3 cos 2 x + cos 3 x$ is

(A)

nπ + \frac{π}{8}

(B)

\frac{nπ}{2} + \frac{π}{8}

(C)

(- 1)^{n} \frac{nπ}{2} + \frac{π}{8}

(D)

2 nπ + cos^{- 1} \frac{3}{2}

JEE Advanced 1981

LEVELBoard

The general solution of the trigonometric equation $sin x + cos x = 1$ is given by :

(A)

x = 2 nπ; n = 0, \pm 1, \pm 2 \dots

(B)

x = 2 nπ + π /2; n = 0, \pm 1, \pm 2 \dots

(C)

x = nπ + (- 1)^{n} \frac{π}{4} - \frac{π}{4}

The number of distinct solutions of the equation $\frac{5}{4} cos^{2} 2 x + cos^{4} x + sin^{4} x + cos^{6} x + sin^{6} x = 2$ in the interval $[0, 2 π]$ is ____.

JEE Advanced 2010

LEVELJEE Main

The number of all possible values of $θ$ where $0 < θ < π$ , for which the system of equations $(y + z) cos 3 θ = (x y z) sin 3 θ$ , $x sin 3 θ = \frac{2 c o s 3 θ}{y} + \frac{2 s i n 3 θ}{z}$ , $(x y z) sin 3 θ = (y + 2 z) cos 3 θ + y sin 3 θ$ have a solution $(x_{0}, y_{0}, z_{0})$ with $y_{0} z_{0} \neq = 0$ , is ____.

JEE Advanced 2010

LEVELJEE Main

The number of values of $θ$ in the interval $(- π /2, π /2)$ such that $θ \neq = \frac{nπ}{5}$ for $n = 0, \pm 1, \pm 2$ and $tan θ = cot 5 θ$ as well as $sin 2 θ = cos 4 θ$ is ____.

JEE Advanced 1997

LEVELJEE Main

The real roots of the equation $cos^{7} x + sin^{4} x = 1$ in the interval $(- π, π)$ are ........., ........., and ..........

JEE Advanced 1996

LEVELJEE Main

General value of $θ$ satisfying the equation $tan^{2} θ + sec 2 θ = 1$ is ..........

JEE Advanced 1996

LEVELJEE Main

Find all values of $θ$ in the interval $(- π /2, π /2)$ satisfying the equation $(1 - tan θ) (1 + tan θ) sec^{2} θ + 2^{t a n^{2} θ} = 0$ .

JEE Advanced 1995

LEVELJEE Main

Find the smallest positive number $p$ for which the equation $cos (p sin x) = sin (p cos x)$ has a solution $x \in [0, 2 π]$ .

JEE Advanced 1987

LEVELBoard

The solution set of the system of equations $x + y = \frac{2 π}{3}$ , $cos x + cos y = \frac{3}{2}$ , where $x$ and $y$ are real, is ..........

JEE Advanced 1987

LEVELJEE Main

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 0≤x<2π, then the number of real values of x, which satisfy the equation cosx+cos2x+cos3x+cos4x=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of values of x in the interval [0,3π] satisfying the equation 2sin2x+5sinx−3=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solution of tanx+secx=2cosx in [0,2π) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={x∈(−π,π):x=0,±π/2}. The sum of all distinct solutions of the equation 3​secx+cscx+2(tanx−cotx)=0 in the set S is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For x∈(0,π), the equation sinx+2sin2x−sin3x=3 has

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let θ,ϕ∈[0,2π] be such that 2cosθ(1−sinϕ)=sin2θ(tan2θ​+cot2θ​)cosϕ−1, tan(2π−θ)>0 and −1<sinθ<−3​/2, then ϕ cannot satisfy

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P={θ:sinθ−cosθ=2​cosθ} and Q={θ:sinθ+cosθ=2​sinθ} be two sets. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solutions of the pair of equations 2sin2θ−cos2θ=0, 2cos2θ−3sinθ=0 in the interval [0,2π] is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The values of θ∈(0,2π) for which 2sin2θ−5sinθ+2>0, are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } cos(α−β)=1 and cos(α+β)=1/e where α,β∈[−π,π]. Pairs of α,β which satisfy both the equations is/are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of values of x in the interval [0,5π] satisfying the equation 3sin2x−7sinx+2=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of values of x where the function f(x)=cosx+cos(2​x) attains its maximum is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The general values of θ satisfying the equation 2sin2θ−sinθ−2=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Number of solutions of the equation tanx+secx=2cosx lying in the interval [0,2π] is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The general solution of sinx−3sin2x+sin3x=cosx−3cos2x+cos3x is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The general solution of the trigonometric equation sinx+cosx=1 is given by :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of distinct solutions of the equation 45​cos22x+cos4x+sin4x+cos6x+sin6x=2 in the interval [0,2π] is ____.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of values of θ in the interval (−π/2,π/2) such that θ=5nπ​ for n=0,±1,±2 and tanθ=cot5θ as well as sin2θ=cos4θ is ____.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The real roots of the equation cos7x+sin4x=1 in the interval (−π,π) are ........., ........., and ..........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } General value of θ satisfying the equation tan2θ+sec2θ=1 is ..........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find all values of θ in the interval (−π/2,π/2) satisfying the equation (1−tanθ)(1+tanθ)sec2θ+2tan2θ=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the smallest positive number p for which the equation cos(psinx)=sin(pcosx) has a solution x∈[0,2π].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The solution set of the system of equations x+y=32π​, cosx+cosy=23​, where x and y are real, is ..........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of all x in the interval [0,π] for which 2sin2x−3sinx+1≥0, is ..........

If $0 \leq x < 2 π$ , then the number of real values of $x$ , which satisfy the equation $cos x + cos 2 x + cos 3 x + cos 4 x = 0$ is

The number of values of $x$ in the interval $[0, 3 π]$ satisfying the equation $2 sin^{2} x + 5 sin x - 3 = 0$ is

The number of solution of $tan x + sec x = 2 cos x$ in $[0, 2 π)$ is

Let $S = {x \in (- π, π) : x \neq = 0, \pm π /2}$ . The sum of all distinct solutions of the equation $3 sec x + csc x + 2 (tan x - cot x) = 0$ in the set $S$ is equal to

For $x \in (0, π)$ , the equation $sin x + 2 sin 2 x - sin 3 x = 3$ has

Let $θ, ϕ \in [0, 2 π]$ be such that $2 cos θ (1 - sin ϕ) = sin^{2} θ (tan \frac{θ}{2} + cot \frac{θ}{2}) cos ϕ - 1$ , $tan (2 π - θ) > 0$ and $- 1 < sin θ < - 3 /2$ , then $ϕ$ cannot satisfy

Let $P = {θ : sin θ - cos θ = 2 cos θ}$ and $Q = {θ : sin θ + cos θ = 2 sin θ}$ be two sets. Then

The number of solutions of the pair of equations $2 sin^{2} θ - cos 2 θ = 0$ , $2 cos^{2} θ - 3 sin θ = 0$ in the interval $[0, 2 π]$ is

The values of $θ \in (0, 2 π)$ for which $2 sin^{2} θ - 5 sin θ + 2 > 0$ , are

$cos (α - β) = 1$ and $cos (α + β) = 1/ e$ where $α, β \in [- π, π]$ . Pairs of $α, β$ which satisfy both the equations is/are

The number of values of $x$ in the interval $[0, 5 π]$ satisfying the equation $3 sin^{2} x - 7 sin x + 2 = 0$ is

The number of values of $x$ where the function $f (x) = cos x + cos (2 x)$ attains its maximum is

The general values of $θ$ satisfying the equation $2 sin^{2} θ - sin θ - 2 = 0$ is

Number of solutions of the equation $tan x + sec x = 2 cos x$ lying in the interval $[0, 2 π]$ is :

The general solution of $sin x - 3 sin 2 x + sin 3 x = cos x - 3 cos 2 x + cos 3 x$ is

The general solution of the trigonometric equation $sin x + cos x = 1$ is given by :

The number of distinct solutions of the equation $\frac{5}{4} cos^{2} 2 x + cos^{4} x + sin^{4} x + cos^{6} x + sin^{6} x = 2$ in the interval $[0, 2 π]$ is ____.

The number of values of $θ$ in the interval $(- π /2, π /2)$ such that $θ \neq = \frac{nπ}{5}$ for $n = 0, \pm 1, \pm 2$ and $tan θ = cot 5 θ$ as well as $sin 2 θ = cos 4 θ$ is ____.

The real roots of the equation $cos^{7} x + sin^{4} x = 1$ in the interval $(- π, π)$ are ........., ........., and ..........

General value of $θ$ satisfying the equation $tan^{2} θ + sec 2 θ = 1$ is ..........

Find all values of $θ$ in the interval $(- π /2, π /2)$ satisfying the equation $(1 - tan θ) (1 + tan θ) sec^{2} θ + 2^{t a n^{2} θ} = 0$ .

Find the smallest positive number $p$ for which the equation $cos (p sin x) = sin (p cos x)$ has a solution $x \in [0, 2 π]$ .

The solution set of the system of equations $x + y = \frac{2 π}{3}$ , $cos x + cos y = \frac{3}{2}$ , where $x$ and $y$ are real, is ..........

The set of all $x$ in the interval $[0, π]$ for which $2 sin^{2} x - 3 sin x + 1 \geq 0$ , is ..........