List of top Mathematics Questions on Trigonometry asked in TS EAMCET

Consider the following statements:
Assertion (A): When $ x, y, z $ are positive numbers, then \[ \tan^{-1}\left( \sqrt{\frac{x(x+y+z)}{yz}} \right) + \tan^{-1}\left( \sqrt{\frac{y(x+y+z)}{xz}} \right) + \tan^{-1}\left( \sqrt{\frac{z(x+y+z)}{xy}} \right) = \pi \] Reason (R): $ \tan^{-1}a + \tan^{-1}b = \tan^{-1}\left( \frac{a+b}{1-ab} \right) $ if $ a>0 $ and $ b>0 $.

In a triangle ABC, if $ r_1=4 $, $ r_2=8 $ and $ r_3=24 $, then $ a:b:c = $

In a triangle ABC, $ (r_2 + r_3)\sec^2\left(\frac{A}{2}\right) = $
(Note: Based on the answer key and standard identities, the function is interpreted as $ \sec^2 $).

If $ 5\sin\theta + 3\cos\left(\theta + \frac{\pi}{3}\right) + 3 $ lies between $ \alpha $ and $ \beta $ (including $ \alpha, \beta $ also), then $ (\alpha-\beta)(\alpha+\beta-6) = $

$ \frac{\sin 1^\circ + \sin 2^\circ + \dots + \sin 89^\circ}{2(\cos 1^\circ + \cos 2^\circ + \dots + \cos 44^\circ) + 1} = $

If $ 3\sin(\alpha-\beta) = 5\cos(\alpha+\beta) $ and $ \alpha+\beta \neq \frac{\pi}{2} $, then $ \frac{\tan(\frac{\pi}{4}-\alpha)}{\tan(\frac{\pi}{4}-\beta)} = $

If $0 \le x \le \frac{3}{4}$, then the number of values of $x$ satisfying the equation $\text{Tan}^{-1}(2x-1) + \text{Tan}^{-1}2x = \text{Tan}^{-1}4x - \text{Tan}^{-1}(2x+1)$ is

Number of solutions of the equation $\sin^2\theta + 2\cos^2\theta - \sqrt{3}\sin\theta\cos\theta = 2$ lying in the interval $(-\pi, \pi)$ is

$16 \sin 12^\circ \cos 18^\circ \sin 48^\circ =$

$\frac{\cos 15^\circ \cos^2 22\frac{1}{2}^\circ - \sin 75^\circ \sin^2 52\frac{1}{2}^\circ}{\cos^2 15^\circ - \cos^2 75^\circ} =$

If $3\sin\theta + 4\cos\theta = 3$ and $\theta \neq (2n+1)\frac{\pi}{2}$, then $\sin 2\theta =$

If $ 0 <\theta <\frac{\pi}{4} $ and $ 8\cos\theta + 15\sin\theta = 15 $, then $ 15\cos\theta - 8\sin\theta = $

Evaluate: \[ \sin 20^\circ (4 + \sec 20^\circ) = ? \]

Suppose $ \theta_1 $ and $ \theta_2 $ are such that $ (\theta_1 - \theta_2) $ lies in the 3rd or 4th quadrant. If \[ \sin\theta_1 + \sin\theta_2 = \frac{21}{65} \quad \text{and} \quad \cos\theta_1 + \cos\theta_2 = \frac{27}{65} \] then \[ \cos\left(\frac{\theta_1 - \theta_2}{2}\right) = \]

If $ A $ is the solution set of the equation $ \cos^2 x = \cos^2 \frac{\pi}{6} $ and $ B $ is the solution set of the equation $ \cos^2 x = \log_{10} P $ where $ P + \frac{16}{P} = 10 $, then $ B - A = ? $

In $ \triangle ABC $, if $ r_1 + r_2 = 3R $, $ r_2 + r_3 = 2R $, then what type of triangle is $ \triangle ABC $?

Let $ \bar{n} $ be a unit vector normal to the plane $ \pi $ containing the vectors $ \bar{T} + 3\bar{k} $ and $ 2\bar{i} + \bar{j} - \bar{k} $. If this plane $ \pi $ passes through the point $ (-3,7,1) $ and $ p $ is the perpendicular distance from the origin to this plane, then $ \sqrt{p^2 + 5} $ is:

If $ \bar{a} = \bar{i} - \bar{j} + 3\bar{k} $, $ \bar{c} = -\bar{k} $ are position vectors of two points and $ \bar{b} = 2\bar{i} - \bar{j} + \lambda \bar{k} $,

$ \bar{d} = \bar{i} + \bar{j} - \bar{k} $ are two vectors, then the lines $ r = \bar{a} + t \bar{b} $, $ r = \bar{c} + s \bar{d} $ are:

If $ \sinh x = \frac{12}{5} $, then $ \sinh 3x + \cosh 3x $ = ?

In a $ \triangle ABC $, the sides $ b, c $ are fixed. In measuring angle $ A $, if there is an error of $ \delta A $, then the percentage error in measuring the length of the side $ a $ is:

If $ m, l, r, s, n $ are integers such that $ 9 >m >l >s >n >r >2 $ and \[ \int_{0}^{\frac{\pi}{2}} \sin^n x \cos^r x \, dx = 4 \int_{0}^{\frac{\pi}{2}} \sin^m x \cos^r x \, dx, \] \[ \int_{0}^{\frac{\pi}{2}} \sin^l x \cos^r x \, dx = 4 \int_{0}^{\frac{\pi}{2}} \sin^s x \cos^r x \, dx, \] \[ \int_{0}^{\frac{\pi}{2}} \sin^n x \cos^r x \, dx = 0, \] then the equation involving $ s, l, m, r $ is:

If $ \triangle ABC $ is an isosceles triangle with base $ BC $, then $ r_1 = ? $

If \[ \int \sqrt{\csc x + 1} \, dx = k \tan^{-1} (f(x)) + c, \] then \[ \frac{1}{k} f\left(\frac{\pi}{6}\right) = ? \]

The trigonometric equation $ \sin^{-1}x = 2\sin^{-1}a $ has a solution. Find the valid range for $ a $.