List of top Mathematics Questions on Trigonometric Equations

The number of elements in the set $\{x\in[0,180^\circ]: \tan(x+100^\circ)=\tan(x+50^\circ)\tan x\tan(x-50^\circ)\}$ is

Let $ \cos(\alpha + \beta) = -\dfrac{1}{10} $ and $ \sin(\alpha - \beta) = \dfrac{3}{8} $, where $ 0<\alpha<\dfrac{\pi}{3} $ and $ 0<\beta<\dfrac{\pi}{4} $.
If \[ \tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, \quad r, s \in \mathbb{N}, \] then the value of $ r + s $ is ______________.

Find the number of solutions of the equation \[ \tan(x+100^\circ)=\tan(x+50^\circ)\tan(x-50^\circ) \] where $x\in(0,\pi)$.

Number of solutions of the equation \[ \sqrt{3} \cos 2\theta + 8 \cos \theta + 3\sqrt{3} = 0 \] in $ \theta \in \left[ -3\pi, 2\pi \right] $:

The value of \[ \frac{\sqrt{3}\,\cosec 20^\circ-\sec 20^\circ}{\cos 20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ} \] is:

If $\cot x=\dfrac{5}{12}$ for some $x\in\left(\pi,\dfrac{3\pi}{2}\right)$, then \[ \sin 7x\left(\cos\frac{13x}{2}+\sin\frac{13x}{2}\right) +\cos 7x\left(\cos\frac{13x}{2}-\sin\frac{13x}{2}\right) \] is equal to:

Find the number of solutions of the equation \[ \tan(x+100^\circ)=\tan(x+50^\circ)\tan(x-50^\circ) \] where $x\in(0,\pi)$.

Principal Solution of $ (5 \sin \theta)(2 \cos \theta + 1) = 0 $

Which of the following expressions represents the Principal Solution

The sum of all values of $ \theta \in [0, 2\pi] $ satisfying $ 2\sin^2\theta = \cos 2\theta $ and $ 2\cos^2\theta = 3\sin\theta $ is:

If $ y = \cos \left( \frac{\pi}{3} + \cos^{-1} \frac{x}{2} \right) $, then $ (x - y)^2 + 3y^2 $ is equal to ________.

If $ \theta \in [-2\pi,\ 2\pi] $, then the number of solutions of $$ 2\sqrt{2} \cos^2\theta + (2 - \sqrt{6}) \cos\theta - \sqrt{3} = 0 $$ is:

The number of solutions of the equation $ \cos 2\theta \cos \left( \frac{\theta}{2} \right) + \cos \left( \frac{5\theta}{2} \right) = 2 \cos^3 \left( \frac{5\theta}{2} \right) $ in the interval $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right ]$ is:

If for $ \theta \in \left[ -\frac{\pi}{3}, 0 \right] $, the points $ (x, y) = \left( 3 \tan\left( \theta + \frac{\pi}{3} \right), 2 \tan\left( \theta + \frac{\pi}{6} \right) \right) $ lie on $ xy + \alpha x + \beta y + \gamma = 0 $, then $ \alpha^2 + \beta^2 + \gamma^2 $ is equal to:

Let $ A = \left\{ \theta \in [0, 2\pi] : \Re\left( \frac{2 \cos \theta + i \sin \theta}{\cos \theta - 3i \sin \theta} \right) = 0 \right\} $. Then $ \sum_{\theta \in A} \theta^2 $ is equal to:

If \[ \lim_{x \to 0} \frac{\cos(2x) + a \cos(4x) - b}{x^4} \] is finite, then $ a + b = $ __.

If $ \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] $, then the number of solutions of the equation \[ \sqrt{3} \csc^2 \theta - 2 (\sqrt{3} - 1) \csc \theta - 4 = 0 \] is:

The sum of all values of $ \theta \in [0, 2\pi] $ satisfying $ 2\sin^2\theta = \cos 2\theta $ and $ 2\cos^2\theta = 3\sin\theta $ is:

If $ 0 \leq a, b \leq 3 $ and the equation $ x^2 + 4 + 3\cos(ax + b) = 2x $ has real solutions, then the value of $ (a + b) $ is:

The solution set of the equation $ \tan(\pi \tan x) = \cot(\pi \cot x) $, $ x \in \left(0, \frac{\pi}{2}\right) $, is:

If $ \cos^{-1}(x) - \sin^{-1}(x) = \frac{\pi}{6} $, then find } $ x $.

The general solution of the equation $ \cos(2x) = \frac{1}{2} $ is:

If $ \sin \theta = \frac{3}{5} $ and $ \theta $ is in the first quadrant, find the value of $ \tan \theta $.

If $ \tan\left( \alpha - \frac{\pi}{12} \right) = \frac{1}{\sqrt{3}} $, find $ \alpha $.

Given $ y = \sec(\tan^{-1}(x)) $, find $ \frac{dy}{dx} $ at $ x = \sqrt{3} $.