Trigonometry PYQ — Page 7

The number of integral values of $k$ for which the equation $3\mathrm{sin}x+4\mathrm{cos}x=k+1$ has a solution, $k\in R$ is _______.

$cosec[2{\mathrm{cot}}^{-1}(5)+{\mathrm{cos}}^{-1}(\frac{4}{5})]$ is equal to:

The number of roots of the equation, ${(81)}^{{\mathrm{sin}}^{2}x}+{(81)}^{{\mathrm{cos}}^{2}x}=30$ in the interval $[0,\pi ]$ is equal to :

If $\mathrm{sin}\theta +\mathrm{cos}\theta =\frac{1}{2},$ then $16(\mathrm{sin}(2\theta )+\mathrm{cos}(4\theta )+\mathrm{sin}(6\theta ))$ is equal to:

The number of real roots of the equation ${\mathrm{tan}}^{-1}\sqrt{x(x+1)}+{\mathrm{sin}}^{-1}\sqrt{{x}^{2}+x+1}=\frac{\pi }{4}$ is:

The value of $2\mathrm{sin}(\frac{\pi }{8})\mathrm{sin}(\frac{2\pi }{8})\mathrm{sin}(\frac{3\pi }{8})\mathrm{sin}(\frac{5\pi }{8})\mathrm{sin}(\frac{6\pi }{8})\mathrm{sin}(\frac{7\pi }{8})$ is :

The value of $\mathrm{tan}(2{\mathrm{tan}}^{-1}(\frac{3}{5})+{\mathrm{sin}}^{-1}(\frac{5}{13}))$ is equal to:

The number of solutions of the equation ${\mathrm{sin}}^{-1}[{x}^{2}+\frac{1}{3}]+{\mathrm{cos}}^{-1}[{x}^{2}-\frac{2}{3}]={x}^{2}$ for $x\in [-1,1]$, and $[x]$ denotes the greatest integer less than or equal to $x,$ is :

$2\pi -({\mathrm{sin}}^{-1}\frac{4}{5}+{\mathrm{sin}}^{-1}\frac{5}{13}+{\mathrm{sin}}^{-1}\frac{16}{65})$ is equal to :

If $y=\sum _{k=1}^{6}k{\mathrm{cos}}^{-1}{\frac{3}{5}\mathrm{cos}kx-\frac{4}{5}\mathrm{sin}kx}$ then $\frac{dy}{dx}$ at $x=0$is

If $S$ is the sum of the first 10 terms of the series, ${\mathrm{tan}}^{-1}(\frac{1}{3})+{\mathrm{tan}}^{-1}(\frac{1}{7})+{\mathrm{tan}}^{-1}(\frac{1}{13})+{\mathrm{tan}}^{-1}(\frac{1}{21})+\ldots \ldots$ then $\mathrm{tan}(S)$ is equal to :

If $\frac{\sqrt{2}sin\alpha }{\sqrt{1+cos2\alpha }}=\frac{1}{7}$ and $\sqrt{\frac{1-cos2\beta }{2}}=\frac{1}{\sqrt{10}},\alpha , \beta \in (0,\frac{\pi }{2})$, then $tan(\alpha +2\beta )$, is equal to

If $L={\mathrm{sin}}^{2}(\frac{\pi }{16})-{\mathrm{sin}}^{2}(\frac{\pi }{8})$ and $M={\mathrm{cos}}^{2}(\frac{\pi }{16})-{\mathrm{sin}}^{2}(\frac{\pi }{8})$

The value of ${\mathrm{cos}}^{3}(\frac{\pi }{8}).\mathrm{cos}(\frac{3\pi }{8})+{\mathrm{sin}}^{3}(\frac{\pi }{8}).\mathrm{sin}(\frac{3\pi }{8})$ is:

If sin A + sin B = 1 and cos A + cos B = 0 then the value of 12cos2A + 4cos2B is:

If the equation ${\mathrm{cos}}^{4}\theta +{\mathrm{sin}}^{4}\theta +\lambda =0$ has real solutions for $\theta$ then $\lambda$ lies in interval

If $0\leq x<\frac{\pi }{2},$ then the number of values of $x$ for which $\mathrm{sin}x-\mathrm{sin}2x+\mathrm{sin}3x=0,$ is:

All $x$ satisfying the inequality $\left(\cot ^{-1} x\right)^{2}-7\left(\cot ^{-1} x\right)+10>$ 0 , lie in the interval :

If $\alpha ={cos}^{-1}(\frac{3}{5})$ , $\beta ={tan}^{-1}(\frac{1}{3})$ , where $0<\alpha ,\beta <\frac{\pi }{2},$ then $\alpha -\beta$ is equal to

Let $S={\theta \in [-2\pi ,2\pi ]:2{\mathrm{cos}}^{2}\theta +3\mathrm{sin}\theta =0}.$ Then the sum of the elements of $S$ is:

The value of ${\mathrm{sin}}^{-1}⁡(\frac{12}{13})-{\mathrm{sin}}^{-1}⁡(\frac{3}{5})$ is equal to:

Let $S$ be the set of all $\alpha \in R$ such that the equation, $cos2x+\alpha sinx=2\alpha -7$ has a solution. Then $S$ is equal to:

For any $\theta \in (\frac{\pi }{4},\frac{\pi }{2}),$ the expression $3{(\mathrm{sin}\theta -\mathrm{cos}\theta )}^{4}+6{(\mathrm{sin}\theta +\mathrm{cos}\theta )}^{2}+4 {sin}^{6}\theta$ equals:

The value of ${\mathrm{cos}}^{2}10^{\circ}–\mathrm{cos}10^{\circ} \mathrm{cos}50^{\circ}+co{s}^{2}50^{\circ}$ is