JEE Advanced Trigonometric Functions PYQ – Solutions of Trigonometric Equations

Q 1 :

Let $S = {x \in (- π, π) : x \neq 0, \pm \frac{π}{2}}$ . The sum of all distinct solutions of the equation $\sqrt{3} \sec x + c o s e c x + 2 (\tan x - \cot x) = 0$ in the set $S$ is equal to [2016]

$- \frac{7 π}{9}$
$- \frac{2 π}{9}$
$0$
$\frac{5 π}{9}$

1

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(3)

$\sqrt{3} \sec x + cosec x + 2 (\tan x - \cot x) = 0$

$\Rightarrow \frac{\sqrt{3}}{2} \sin x + \frac{1}{2} \cos x = \cos^{2} x - \sin^{2} x$

$\Rightarrow \cos (x - \frac{π}{3}) = \cos 2 x$ $\Rightarrow x - \frac{π}{3} = 2 n π \pm 2 x$

$\Rightarrow x = \frac{2 n π}{3} + \frac{π}{9} or x = - 2 n π - \frac{π}{3}$

$For x \in S, n = 0 \Rightarrow x = \frac{π}{9}, - \frac{π}{3}$

$Now, n = 1 \Rightarrow x = \frac{7 π}{9} and n = - 1 \Rightarrow x = - \frac{5 π}{9}$

$Hence, sum of all values of x = \frac{π}{9} - \frac{π}{3} + \frac{7 π}{9} - \frac{5 π}{9} = 0$

Q 2 :

For $x \in (0, π),$ the equation $\sin x + 2 \sin 2 x - \sin 3 x = 3$ has [2014]

infinitely many solutions
three solutions
one solution
no solution

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4

(4)

$\sin x + 2 \sin 2 x - \sin 3 x = 3$

$\Rightarrow \sin x + 4 \sin x \cos x - 3 \sin x + 4 \sin^{3} x = 3$

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

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

$2 + 4 \cos x - 4 \cos^{2} x = \frac{3}{\sin x}$ $[∵ 0 \leq \sin x \leq 1]$

$\Rightarrow 2 - 4 (\cos^{2} x - 2 \cos x \cdot \frac{1}{2} + \frac{1}{4}) + 1 = \frac{3}{\sin x}$

$\Rightarrow 3 - 4 {(\cos x - \frac{1}{2})}^{2} = \frac{3}{\sin x}$ $∴$ $L.H.S \leq 3 and R.H.S \geq 3$

$Hence, the equation has no solution.$

Q 3 :

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 [2007]

zero
one
two
four

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(3)

$2 \sin^{2} θ - \cos 2 θ = 0 \Rightarrow 1 - 2 \cos 2 θ = 0$

$\Rightarrow \cos 2 θ = \frac{1}{2} \Rightarrow 2 θ = \frac{π}{3}, \frac{5 π}{3}, \frac{7 π}{3}, \frac{11 π}{3}$

$\Rightarrow θ = \frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, \frac{11 π}{6} . . . (i)$

$where θ \in [0, 2 π]$

$Also, 2 \cos^{2} θ - 3 \sin θ = 0$

$\Rightarrow 2 \sin^{2} θ + 3 \sin θ - 2 = 0$

$\Rightarrow (2 \sin θ - 1) (\sin θ + 2) = 0 \Rightarrow \sin θ = \frac{1}{2}$ $[∵ \sin θ \neq - 2]$

$\Rightarrow θ = \frac{π}{6}, \frac{5 π}{6} . . . (i i)$

$where θ \in [0, 2 π]$

$Combining (i) and (ii), we get θ = \frac{π}{6}, \frac{5 π}{6}$

$Hence, there are two solutions.$

Q 4 :

$\cos (α - β) = 1$ and $\cos (α + β) = \frac{1}{e},$ where $α, β \in [- π, π]$ . Pairs of $α, β$ which satisfy both the equations is/are [2005]

0
1
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4

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4

(4)

$Given: \cos (α - β) = 1 and \cos (α + β) = \frac{1}{e}, where α, β \in [- π, π]$

$Now, \cos (α - β) = 1 \Rightarrow α - β = 0 \Rightarrow α = β$

$and \cos (α + β) = \frac{1}{e} \Rightarrow \cos 2 α = \frac{1}{e}$

$∵$ $0 < \frac{1}{e} < 1$

$Now 2 α \in [- 2 π, 2 π]$

$\Rightarrow There will be two values of 2 α in [- 2 π, 0] satisfying \cos 2 α = \frac{1}{e} and two values in [0, 2 π] .$

$\Rightarrow There will be four values of α in [- π, π] and correspondingly four values of β . Hence there are four sets of (α, β) .$

Q 5 :

The number of integral values of $k$ for which the equation $7 \cos x + 5 \sin x = 2 k + 1$ has a solution is [2002]

4
8
10
12

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(2)

$We know, - \sqrt{a^{2} + b^{2}} \leq a \cos θ + b \sin θ \leq \sqrt{a^{2} + b^{2}}$

$\Rightarrow - \sqrt{74} \leq 7 \cos x + 5 \sin x \leq \sqrt{74}$

$\Rightarrow - \sqrt{74} \leq 2 k + 1 \leq \sqrt{74} \Rightarrow - 8.6 \leq 2 k + 1 \leq 8.6$

$\Rightarrow - 4.8 \leq k \leq 3.8$

$Hence, k can take only 8 integral values.$

Q 6 :

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 [2015]

0%

(8)

$\frac{5}{4} \cos^{2} 2 x + \cos^{4} x + \sin^{4} x + \cos^{6} x + \sin^{6} x = 2$

$\Rightarrow \frac{5}{4} \cos^{2} 2 x + 1 - \frac{1}{2} \sin^{2} 2 x + 1 - \frac{3}{4} \sin^{2} 2 x = 2$

$\Rightarrow \frac{5}{4} (\cos^{2} 2 x - \sin^{2} 2 x) = 0 \Rightarrow \cos 4 x = 0$

$\Rightarrow 4 x = (2 n + 1) \frac{π}{2} or x = (2 n + 1) \frac{π}{8}$

$For x \in [0, 2 π], n can take values 0 to 7$

$Hence, there are 8 solutions.$

Q 7 :

The positive integer value of $n > 3$ satisfying the equation $\frac{1}{\sin (\frac{π}{n})} = \frac{1}{\sin (\frac{2 π}{n})} + \frac{1}{\sin (\frac{3 π}{n})}$ is [2011]

0%

(7)

$\frac{1}{\sin \frac{π}{n}} - \frac{1}{\sin \frac{3 π}{n}} = \frac{1}{\sin \frac{2 π}{n}}$

$\Rightarrow \frac{\sin \frac{3 π}{n} - \sin \frac{π}{n}}{\sin \frac{π}{n} \sin \frac{3 π}{n}} = \frac{1}{\sin \frac{2 π}{n}}$ $\Rightarrow \frac{2 \cos \frac{2 π}{n} \sin \frac{π}{n}}{\sin \frac{π}{n} \sin \frac{3 π}{n}} = \frac{1}{\sin \frac{2 π}{n}}$

$\Rightarrow 2 \sin \frac{2 π}{n} \cos \frac{2 π}{n} = \sin \frac{3 π}{n}$ $\Rightarrow \sin \frac{4 π}{n} - \sin \frac{3 π}{n} = 0$

$\Rightarrow 2 \cos \frac{7 π}{2 n} \sin \frac{π}{2 n} = 0 \Rightarrow \cos \frac{7 π}{2 n} = 0 or \sin \frac{π}{2 n} = 0$

$\Rightarrow \frac{7 π}{2 n} = (2 k + 1) \frac{π}{2} or \frac{π}{2 n} = 2 k π, where k \in ℤ$

$\Rightarrow n = \frac{7}{2 k + 1} or n = \frac{1}{4 k}$

$(n = \frac{1}{4 k} not possible for any integral value of k)$

$As n > 3; for k = 0, we get n = 7 .$

Q 8 :

Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3} + 1$ apart. If the chords subtend at the center, angles of $\frac{π}{k}$ and $\frac{2 π}{k}$ , where $k > 0$ , then the value of $[k]$ is

[Note: $[k]$ denotes the largest integer less than or equal to $k$ ] [2010]

0%

(3)

From the figure,

[IMAGE 4]

$2 \cos \frac{π}{k} + 2 \cos \frac{π}{2 k} = \sqrt{3} + 1$

$\Rightarrow 2 \times 2 \cos^{2} \frac{π}{2 k} + 2 \cos \frac{π}{2 k} - 2 = \sqrt{3} + 1$

$\Rightarrow 4 \cos^{2} \frac{π}{2 k} + 2 \cos \frac{π}{2 k} - (3 + \sqrt{3}) = 0$

$\Rightarrow \cos \frac{π}{2 k} = \frac{- 2 \pm \sqrt{4 + 16 (3 + \sqrt{3})}}{8}$ $= \frac{- 1 \pm \sqrt{13 + 4 \sqrt{3}}}{4}$

$= \frac{- 1 \pm (2 \sqrt{3} + 1)}{4}$ $= \frac{\sqrt{3}}{2} or - (\frac{\sqrt{3} + 1}{2})$

$As \frac{π}{2 k} is an acute angle, \cos \frac{π}{2 k} = \frac{\sqrt{3}}{2} = \cos \frac{π}{6}$ $\Rightarrow k = 3$

Q 9 :

The number of values of $θ$ in the interval $(- \frac{π}{2}, \frac{π}{2})$ such that $θ \neq \frac{n π}{5}$ for $n = 0, \pm 1, \pm 2$ and $\tan θ = \cot 5 θ$ as well as $\sin 2 θ = \cos 4 θ$ is [2010]

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(3)

$\tan θ = \cot 5 θ, θ \neq \frac{n π}{5}$

$\Rightarrow \cos θ \cos 5 θ - \sin 5 θ \sin θ = 0 \Rightarrow \cos 6 θ = 0$

$\Rightarrow 6 θ = \frac{- 5 π}{2}, \frac{- 3 π}{2}, \frac{- π}{2}, \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}$

$\Rightarrow θ = \frac{- 5 π}{12}, \frac{- π}{4}, \frac{- π}{12}, \frac{π}{12}, \frac{π}{4}, \frac{5 π}{12}$

$Again \sin 2 θ = \cos 4 θ = 1 - 2 \sin^{2} 2 θ$

$\Rightarrow 2 \sin^{2} 2 θ + \sin 2 θ - 1 = 0 \Rightarrow \sin 2 θ = - 1, \frac{1}{2}$

$\Rightarrow 2 θ = \frac{- π}{2}, \frac{π}{6}, \frac{5 π}{6} \Rightarrow θ = \frac{- π}{4}, \frac{π}{12}, \frac{5 π}{12}$

$So, common solutions are θ = \frac{- π}{4}, \frac{π}{12}, \frac{5 π}{12}$

$∴$ $Number of solutions = 3$

Q 10 :

The number of all possible values of $θ$ where $0 < θ < π,$ for which the system of equations

$(y + z) \cos 3 θ = (x y z) \sin 3 θ$

$x \sin 3 θ = \frac{2 \cos 3 θ}{y} + \frac{2 \sin 3 θ}{z}$

$(x y z) \sin 3 θ = (y + 2 z) \cos 3 θ + y \sin 3 θ$

$have a solution (x_{0}, y_{0}, z_{0}) with y_{0} z_{0} \neq 0, is.$ [2010]

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(3)

Given equations are

$x y z \sin 3 θ = (y + z) \cos 3 θ . . . (i)$

$x y z \sin 3 θ = 2 z \cos 3 θ + 2 y \sin 3 θ . . . (i i)$

$x y z \sin 3 θ = (y + 2 z) \cos 3 θ + y \sin 3 θ . . . (i i i)$

$On subtracting eq. (ii) from (i), we get$

$(\cos 3 θ - 2 \sin 3 θ) y - (\cos 3 θ) z = 0 . . . (i v)$

$On subtracting eq. (i) from (iii), we get$

$\sin 3 θ y + (\cos 3 θ) z = 0 . . . (v)$

$Eq. (iv) and (v) form a homogeneous system of linear equations.$

$But y \neq 0, z \neq 0$

$∴$ $\frac{\cos 3 θ - 2 \sin 3 θ}{\sin 3 θ} = - \frac{\cos 3 θ}{\cos 3 θ} \Rightarrow \cos 3 θ = \sin 3 θ$

$\Rightarrow \tan 3 θ = 1 \Rightarrow 3 θ = n π + \frac{π}{4}$ $\Rightarrow θ = (4 n + 1) \frac{π}{12}, n \in ℤ$

$For θ \in (0, π) \Rightarrow θ = \frac{π}{12}, \frac{5 π}{12}, \frac{3 π}{4}$

$∴$ $Three such solutions are possible.$

Topic Question Set

Q 1 :

Let $S = {x \in (- π, π) : x \neq 0, \pm \frac{π}{2}}$ . The sum of all distinct solutions of the equation $\sqrt{3} \sec x + c o s e c x + 2 (\tan x - \cot x) = 0$ in the set $S$ is equal to [2016]

Q 2 :

For $x \in (0, π),$ the equation $\sin x + 2 \sin 2 x - \sin 3 x = 3$ has [2014]

Q 3 :

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 [2007]

Q 4 :

$\cos (α - β) = 1$ and $\cos (α + β) = \frac{1}{e},$ where $α, β \in [- π, π]$ . Pairs of $α, β$ which satisfy both the equations is/are [2005]

Q 5 :

The number of integral values of $k$ for which the equation $7 \cos x + 5 \sin x = 2 k + 1$ has a solution is [2002]

Q 6 :

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 [2015]

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Q 7 :

The positive integer value of $n > 3$ satisfying the equation $\frac{1}{\sin (\frac{π}{n})} = \frac{1}{\sin (\frac{2 π}{n})} + \frac{1}{\sin (\frac{3 π}{n})}$ is [2011]

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Q 8 :

Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3} + 1$ apart. If the chords subtend at the center, angles of $\frac{π}{k}$ and $\frac{2 π}{k}$ , where $k > 0$ , then the value of $[k]$ is

[Note: $[k]$ denotes the largest integer less than or equal to $k$ ] [2010]

0%

Q 9 :

The number of values of $θ$ in the interval $(- \frac{π}{2}, \frac{π}{2})$ such that $θ \neq \frac{n π}{5}$ for $n = 0, \pm 1, \pm 2$ and $\tan θ = \cot 5 θ$ as well as $\sin 2 θ = \cos 4 θ$ is [2010]

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Topic Question Set

Q 1 : Let S={x∈(-π,π):x≠0,±π2}. The sum of all distinct solutions of the equation 3secx+cosec x+2(tanx-cotx)=0 in the set S is equal to [2016]

Q 2 : For x∈(0,π), the equation sinx+2sin 2x-sin 3x=3 has [2014]

Q 3 : The number of solutions of the pair of equations 2sin2θ-cos2θ=0 2cos2θ-3sinθ=0 in the interval [0,2π] is [2007]

Q 4 : cos(α-β)=1 and cos(α+β)=1e, where α,β∈[-π,π]. Pairs of α,β which satisfy both the equations is/are [2005]

Q 5 : The number of integral values of k for which the equation 7cosx+5sinx=2k+1 has a solution is [2002]

Q 6 : The number of distinct solutions of the equation 54cos22x+cos4x+sin4x+cos6x+sin6x=2 in the interval [0,2π] is [2015]

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Q 7 : The positive integer value of n>3 satisfying the equation 1sin(πn)=1sin(2πn)+1sin(3πn) is [2011]

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Q 8 : Two parallel chords of a circle of radius 2 are at a distance 3+1 apart. If the chords subtend at the center, angles of πk and 2πk, where k>0, then the value of [k] is [Note: [k] denotes the largest integer less than or equal to k] [2010]

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Q 9 : The number of values of θ in the interval (-π2,π2) such that θ≠nπ5 for n=0,±1,±2 and tanθ=cot5θ as well as sin2θ=cos4θ is [2010]

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Q 10 : The number of all possible values of θ where 0<θ<π, for which the system of equations (y+z)cos3θ=(xyz)sin3θ xsin3θ=2cos3θy+2sin3θz (xyz)sin3θ=(y+2z)cos3θ+ysin3θ have a solution (x0,y0,z0) with y0z0≠0, is. [2010]

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Q 1 :

Let $S = {x \in (- π, π) : x \neq 0, \pm \frac{π}{2}}$ . The sum of all distinct solutions of the equation $\sqrt{3} \sec x + c o s e c x + 2 (\tan x - \cot x) = 0$ in the set $S$ is equal to [2016]

Q 2 :

For $x \in (0, π),$ the equation $\sin x + 2 \sin 2 x - \sin 3 x = 3$ has [2014]

Q 3 :

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 [2007]

Q 4 :

$\cos (α - β) = 1$ and $\cos (α + β) = \frac{1}{e},$ where $α, β \in [- π, π]$ . Pairs of $α, β$ which satisfy both the equations is/are [2005]

Q 5 :

The number of integral values of $k$ for which the equation $7 \cos x + 5 \sin x = 2 k + 1$ has a solution is [2002]

Q 6 :

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 [2015]

Q 7 :

The positive integer value of $n > 3$ satisfying the equation $\frac{1}{\sin (\frac{π}{n})} = \frac{1}{\sin (\frac{2 π}{n})} + \frac{1}{\sin (\frac{3 π}{n})}$ is [2011]

Q 8 :

Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3} + 1$ apart. If the chords subtend at the center, angles of $\frac{π}{k}$ and $\frac{2 π}{k}$ , where $k > 0$ , then the value of $[k]$ is

[Note: $[k]$ denotes the largest integer less than or equal to $k$ ] [2010]

Q 9 :

The number of values of $θ$ in the interval $(- \frac{π}{2}, \frac{π}{2})$ such that $θ \neq \frac{n π}{5}$ for $n = 0, \pm 1, \pm 2$ and $\tan θ = \cot 5 θ$ as well as $\sin 2 θ = \cos 4 θ$ is [2010]