Basic Inverse Trigonometric Functions

Q.

If $1+{\sin }^2\left(\mathrm{\theta }\right)=3\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right),$ then prove that $\tan \left(\mathrm{\theta }\right)=1$ or $\frac{1}{2}$.

Q.

If $\frac{\left({\sin }^{-1}x\right)}{a}=\frac{\left({\cos }^{-1}x\right)}{b}=\frac{\left({\tan }^{-1}y\right)}{c};0<x<1$, then the value of $\cos \left(\frac{\pi c}{a+b}\right)$is:

1. $\frac{1-y^2}{2y}$

2. $\frac{1-y^2}{1+y^2}$

3. $1-y^2$

4. $\frac{1-y^2}{y\sqrt{y}}$

Q.

The sum of possible values of $x$ for ${\tan }^{-1}(x+1)+co{t}^{-1}\frac{1}{(x-1)}={\tan }^{-1}\frac{8}{31}$ is

1. $\frac{–33}{4}$

2. $\frac{–32}{4}$

3. $\frac{–31}{4}$

4. $\frac{–30}{4}$

Q.

Evaluate ${\tan }^{-1}\left[\frac{\cos \mathrm{x}}{1+\sin \mathrm{x}}\right]$

1. $\left(\frac{\mathrm{\pi }}{4}\right)-\left(\frac{\mathrm{x}}{2}\right)$

2. $\left(\frac{\mathrm{\pi }}{4}\right)+\left(\frac{\mathrm{x}}{2}\right)$

3. $\frac{\mathrm{x}}{2}$

4. $\left(\frac{\mathrm{\pi }}{4}\right)-\mathrm{x}$

Q.

${\cos }^4\theta -{\sin }^4\theta$ is equal to

1. $1+2{\sin }^2\left(\frac{\theta }{2}\right)$

2. $2{\cos }^2\theta –1$

3. $1–2{\sin }^2\left(\frac{\theta }{2}\right)$

4. $1+2{\cos }^2\theta$

Q.

$co{t}^{-1}\left[{(\cos \alpha )}^{½}\right]-{\tan }^{-1}\left[{(\cos \alpha )}^{½}\right]=x$, then $\sin x=$

1. ${\tan }^2\left(\frac{\alpha }{2}\right)$

2. $co{t}^2\left(\frac{\alpha }{2}\right)$

3. $\tan \alpha$

4. $cot\frac{\alpha }{2}$

Q. Find the range of f(x)= sgn(2^x)+sgn(|x+5|)

Q. The function y = cos x is one - to - one and onto in the interval $[\frac{-\pi }{2},\frac{\pi }{2}]$

1. True
2. False

Q.

$2{\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{1}{2}\right)=$

1. $90°$

2. $60°$

3. $45°$

4. ${\tan }^{-1}2$

Q.

Find the value of cos ${225}^{\circ }$

Q. The range of f(x) = log to the base root5 (root 2(sinx-cosx)+3) is?

Q. The domain of the function $y={\sin }^{-1}(-x^2)$ is

1. $[0,1]$
2. $(0,1)$
3. $[-1,1]$
4. $(-\infty ,-1)\cup (1,\infty )$

Q.

Prove that $\sqrt{(se{c}^2\theta +\cos e{c}^2\theta )}=\tan \theta +cot\theta$

Q.

${(\cos \alpha +\cos \beta )}^2+{(\sin \alpha +\sin \beta )}^2=?$

1. $4{\cos }^2\frac{(\alpha –\beta )}{2}$

2. $4{\sin }^2\frac{(\alpha –\beta )}{2}$

3. $4{\cos }^2\frac{(\alpha +\beta )}{2}$

4. $4{\sin }^2\frac{(\alpha +\beta )}{2}$

Q. If ${\cot }^{-1}\left(\frac{n}{\pi }\right)>\frac{\pi }{6},n\in N,$ then the maximum value of $n$ is

1. $1$
2. $6$
3. $5$
4. $3$

Q.

Prove that $(cosec\theta -sin\theta )(sec\theta -cos\theta )=\frac{1}{tan\theta +cot\theta }$

Q.

Prove

sinA.sin(60-A).sin(60+A)=sin3A/4

Q.

$\sin \left[{\cot }^{-1}\left(\cos {\tan }^{-1}x\right)\right]=$

1. $\frac{x}{\sqrt{x^2+2}}$

2. $\frac{x}{\sqrt{x^2+1}}$

3. $\frac{1}{\sqrt{x^2+2}}$

4. $\sqrt{\frac{x^2+1}{x^2+2}}$

Q.

How do you prove $\sin 3x=3\sin x-4{\sin }^{3}x$?

Q.

In a $∆ABC$, let $\angle C=\frac{\pi }{2}$, if $r$ is the inradius and $R$ is the circumradius of the $∆ABC$, then $2(r+R)$ equals

1. $c+a$

2. $a+b+c$

3. $a+b$

4. $b+c$

Q.

If ${\sin }^{2}x+2\cos y+xy=0$, then $\frac{dy}{dx}$

1. $\frac{(y+2\sin x)}{(2\sin y+x)}$

2. $\frac{(y+\sin 2x)}{(2\sin y-x)}$

3. $\frac{(y+\sin 2x)}{(\sin y+x)}$

4. none of these

Q.

How do you find the integral of $\int \left(\sin x.\tan x\right)dx$ ?

Q.

If ${\tan }^{-1}\left(\frac{a}{x}\right)+{\tan }^{-1}\left(\frac{b}{x}\right)=\frac{\pi }{2}$, then $x=$

1. $\sqrt{ab}$

2. $\sqrt{2ab}$

3. $2ab$

4. $ab$

Q. The value of $\underset{n\to \infty }{lim}\frac{1}{n}\sum _{j=1}^n\frac{(2j-1)+8n}{(2j-1)+4n}$ is equal to

1. $1+2{\log }_e\left(\frac{3}{2}\right)$
2. $2-{\log }_e\left(\frac{2}{3}\right)$
3. $3+2{\log }_e\left(\frac{2}{3}\right)$
4. $5+{\log }_e\left(\frac{3}{2}\right)$

Q. Let $f\left(x\right)=\cos \left({\cos }^{-1}\left(\frac{2}{\left[x\right]-1}\right)\right)$ where $[\cdot ]$ is the greatest integer function, then

1. Domain of $f\left(x\right)=R-\{0,1\}$
2. Range of $f\left(x\right)=[-1,1]-\left\{0\right\}$
3. Domain of $f\left(x\right)$ excludes $3$ integral values of $x$
4. Range of $f\left(x\right)=\{\frac{2}{k}:k\in Z-\{-1,0,1\left\}\right\}$

Q. The curve y = f(x) is such that the area of the trapezium formed by the coordinate axes, ordinate of an arbitrary point and the tangent at this point equals half the square of its abscissa. The equation of the curve can be

1. $y=c{x}^2\pm x$
2. $y=c{x}^2\pm 1$
3. $y=cx\pm x^2$
4. $y=c{x}^2\pm x^2\pm 1$

Q. If $x,y\in [-1,1],$ $\cos \left({\sin }^{-1}x\right)+\sin \left({\cos }^{-1}y\right)=\frac{\sqrt{11}}{2},$ $(1-x^2)(1-y^2)=\frac{4}{9},$ and $x^2+y^2=\frac{a}{b^2+3}$ for positive integers $a$ and $b,$ then the value of $a+b$ is

1. $14$
2. $7$
3. $21$
4. $10$

Q. The three sides of a right-angled triangle are in G.P. (geometrical progression). If the two acute angles be $\alpha$ and $\beta$, then $\tan \alpha$ and $\tan \beta$ are

1. $\frac{\sqrt{5}+1}{2}$ and $\frac{\sqrt{5}-1}{2}$
2. $\sqrt{\frac{\sqrt{5}+1}{2}}$ and $\sqrt{\frac{\sqrt{5}-1}{2}}$
3. $\sqrt{5}$ and $\frac{1}{\sqrt{5}}$
4. $\frac{\sqrt{5}}{2}$ and $\frac{2}{\sqrt{5}}$

Q. For positive interger $n,$ if $f\left(n\right)={\sin }^n\theta +{\cos }^n\theta ,$ then $\frac{f\left(3\right)-f\left(5\right)}{f\left(5\right)-f\left(7\right)}$ is

1. $\frac{f\left(1\right)}{f\left(3\right)}$
2. $\frac{f\left(3\right)}{f\left(1\right)}$
3. $\frac{f\left(3\right)}{f\left(5\right)}$
4. $\frac{f\left(5\right)}{f\left(7\right)}$

Q.

If sec $\theta$ is the eccentricity of a hyperbola then the eccentricity of the conjugate hyberpola is

1. tan θ

2. cot θ

3. cos θ

4. cosec θ