Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

HARD

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Solve for $x$ :

$\tan x + \tan 2 x + \sqrt{3} \tan x \tan 2 x = \sqrt{3}$ .

Important Questions on Trigonometric Equations and Inequations

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Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

The number of principal solutions of

\tan 2 θ = 1

is:

MEDIUM

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

The equation

\sqrt{3} \sin x + \cos x = 4

has

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

If

5 (\tan^{2} x - \cos^{2} x) = 2 \cos 2 x + 9,

then the value of

\cos 4 x

is

MEDIUM

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

The number of values of $α$ in $[0, 2 π]$ for which $2 \sin^{3} α - 7 \sin^{2} α + 7 sinα = 2$ , is :

MEDIUM

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

Let

a, b, c

be three non-zero real numbers such that the equation

\sqrt{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{a}{b}

is

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

The angles

α, β, γ

of a triangle satisfy the equations

2 \sin α + 3 \cos β = 3 \sqrt{2} a n d 3 \sin β + 2 \cos α = 1

. Then angle

γ

equals

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

Let

X = {x \in R : \cos (\sin x) = \sin (\cos x)} .

The number of elements in

X

is

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

The number of

x \in [0, 2 π]

for which

|\sqrt{2 \sin^{4} x + 18 \cos^{2} x} - \sqrt{2 \cos^{4} x + 18 \sin^{2} x}| = 1

is:

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

One of the solutions of the equation

8 \sin^{3} θ - 7 + \sin θ + \sqrt{3} \cos θ = 0

lies in the interval

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

The number of real solutions of the equation

2 s i n 3 x + \sin 7 x - 3 = 0

which lie in the interval

[- 2 π, 2 π]

is

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

If sum of all the solutions of the equation

8 \cos x \cdot (\cos (\frac{π}{6} + x) \cdot \cos (\frac{π}{6} - x) - \frac{1}{2}) = 1

in

[0, π]

is

k π

, then

k

is equal to:

MEDIUM

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

The sum of all values of

θ \in (0, \frac{π}{2})

satisfying

\sin^{2} 2 θ + \cos^{4} 2 θ = \frac{3}{4}

is

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

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

EASY

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

The number of solutions to the equation

\cos^{4} x + \frac{1}{\cos^{2} x} = \sin^{4} x + \frac{1}{\sin^{2} x}

in the interval

[0, 2 π]

is

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

For

x \in (0, π),

the equation

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

has

MEDIUM

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

Let

S = \{θ \in [- 2 π, 2 π] : 2 \cos^{2} θ + 3 \sin θ = 0\} .

Then the sum of the elements of

S

is:

MEDIUM

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

If

m

and

M

are the minimum and the maximum values of

4 + \frac{1}{2} \sin^{2} 2 x - 2 \cos^{4} x, x \in R,

then

M - m

is equal to:

EASY

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

The general solution of

\sqrt{3} \cos x + \sin x = \sqrt{2},

for an integer

n

is

EASY

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

If

0 \leq x < \frac{π}{2},

then the number of values of

x

for which

\sin x - \sin 2 x + \sin 3 x = 0,

is:

HARD

Mathematics>Trigonometry>Trigonometric Equations and Inequations>Trigonometric Equations

Let

S = \{x; x \in (- π, π) : x \neq 0, \pm \frac{π}{2}\}

. The sum of all distinct solutions of the equation

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

in the set

S

is equal to