Range of Trigonometric Expressions

Q. Find the principal value of ${\tan }^{-1}(-\sqrt{3})$

Q. Find the principal value of ${\tan }^{-1}(-1).$

Q.

The range of $f\left(x\right)=\frac{1}{1-2cosx}$ is

1. $[\frac{1}{3},1]$

2. $[-1,\frac{1}{3}]$

3. $[\frac{-1}{3},1]$

4. $[-\infty ,-1]\cup [\frac{1}{3},\infty ]$

Q.

If $u={\sin }^{-1}\left(\frac{y}{x}\right),$ then $\frac{\partial u}{\partial x}$ is equal to

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

2. $\frac{x}{\surd (1–y^2)}$

3. $\frac{-y}{\surd (x^2–y^2)}$

4. $\frac{-y}{x\surd (x^2–y^2)}$

Q. The domain and range of the function $f\left(x\right)=\frac{1}{2-\cos 3x}$ is

1. $R-\left\{2\right\},[\frac{1}{3},1]$
2. $R,[\frac{1}{3},1]$
3. $R,[\frac{-1}{3},\frac{1}{3}]$
   
4. $R-\left\{2\right\},[\frac{-1}{3},\frac{1}{3}]$

Q.

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

1. $(\frac{-1}{2},\frac{-1}{4}]$

2. $\left[-1,\frac{-1}{2}\right]$

3. $\left[\frac{-3}{2},\frac{-5}{4}\right]$

4. $\left(\frac{-5}{4},-1\right)$

Q. The number of integral value(s) in the range of the expression $\frac{6\tan x+4-4{\tan }^{2}x}{1+{\tan }^{2}x}$ is

Q.

Write the maximum and minimum values of cos (cos x).

Q.

Write the maximum and minimum values of sin (sin x).

Q. Let $x,y,z$ be real numbers with $x\ge y\ge z\ge \frac{\pi }{12}$ such that $x+y+z=\frac{\pi }{2}$. If $p=\cos x\sin y\cos z,$ then

1. the maximum value of $p$ is $\frac{2+\sqrt{3}}{8}$
2. the minimum value of $p$ is $\frac{1}{8}$
3. the maximum value of $p$ is attained when $x=y=\frac{5\pi }{24}$ and $z=\frac{\pi }{12}$
4. the minimum value of $p$ is attained when $x=y=z=\frac{\pi }{6}$

Q.

The minimum value of $f\left(x\right)={\sin }^4\left(x\right)+{\cos }^4\left(x\right)$ , $0\le x\le \frac{\mathrm{\pi }}{2}$ is

1. $\frac{1}{2\sqrt{2}}$

2. $\frac{1}{4}$

3. $-\frac{1}{2}$

4. $\frac{1}{2}$

Q.

Write the maximum value of sin (cos x).

Q. The maximum value of $3\cos \theta +5\sin (\theta -\frac{\pi }{6})$ for any real value of $\theta$ is:

1. $\frac{\sqrt{79}}{2}$
2. $\sqrt{19}$
3. $\sqrt{31}$
4. $\sqrt{34}$

Q.

The range of $f\left(x\right)=cos\left[x\right]$, for $-\frac{\pi }{2}<x<\frac{\pi }{2}$ is

1. [-1, 1]

2. {cos 1, - cos 1, 1}

3. {cos 1, cos 2, 1}

4. {-1, 1, 0}

Q. 36. Differentiate the function with respect to x :- Sin(ax+b)/cos(cx+d)

Q.

$Generalsolutionoftan5\theta =cot2\theta is$

1. $\frac{n\pi }{7}+\frac{\pi }{2},n\in Z$

2. $\theta =\frac{n\pi }{7}+\frac{\pi }{3},n\in Z$

3. $\theta =\frac{n\pi }{7}+\frac{\pi }{14},n\in Z$

4. $\theta =\frac{n\pi }{7}-\frac{\pi }{14},n\in Z$

Q.

Function $f\left(x\right)=\frac{[\lambda \sin x+6\cos x]}{[2\sin x+3\cos x]}$ is monotonic increasing, if

1. $\lambda >1$

2. $\lambda <1$

3. $\lambda <4$

4. $\lambda >4$

Q. The range of the function $f\left(x\right)=\frac{1}{3\sin x+4\cos x-2}$ is

1. $R-\left\{0\right\}$
2. $[-\frac{1}{7},\frac{1}{3}]$
3. $R-[-\frac{1}{7},\frac{1}{3}]$
4. $R-(-\frac{1}{7},\frac{1}{3})$

Q. If $x$ and $y$ are positive integers satisfying ${\tan }^{-1}\left(\frac{1}{x}\right)+{\tan }^{-1}\left(\frac{1}{y}\right)={\tan }^{-1}\left(\frac{1}{7}\right),$ then the number of ordered pairs of $(x,y)$ is

Q. Prove that $(2\sqrt{3}+3)sin\theta +2\sqrt{3}cos\theta$ lies between -$(2\sqrt{3}+\sqrt{15})$ and $(2\sqrt{3}+\sqrt{15})$.

Q. The minimum value of $({\sin }^2\theta +{\cos }^2\theta +{\sec }^2\theta +{\text{cosec}}^2\theta +{\tan }^2\theta +{\cot }^2\theta )$ is

Q.

If in a $\Delta ABC$, tan A + tan B + tan C = 0, then cot A cot B cot C =

1. $\frac{1}{6}$

2. 6

3. 1

4. none of these

Q. ${lim}_{h\to 0}\frac{(a+h{)}^{2}sin(a+h)-a^{2}sina}{h}=$

1. $acosa+a^{2}sina$
2. $asina+a^{2}cosa$
3. $2asina+a^{2}cosa$
4. $2acosa+a^{2}sina$

Q. Let $f$ be a function defined as $f\left(x\right)={\sec }^{-1}(1+{\cos }^{2}x),$ then

1. the domain of $f$ is $R$
2. the domain of $f$ is $[0,\pi ]$
3. the range of $f$ is $[0,\frac{\pi }{2}]$
4. the range of $f$ is $[0,\frac{\pi }{3}]$

Q. The general solution of $ta{n}^2\theta =3is\left(nϵZ\right)$

1. 
2. 
3. 
4. 

Q.

The value of a cos $\theta$ + b sin $\theta$ lies between

1. a and b

2. a-b and a+b

3. 

4. 

Q. If $I=\int e^{x}sin2xdx$, then for what value of K, $KI=e^x(sin2x-2cos2x)+constant$

1. 7
2. 1
3. 3
4. 5

Q.

If $y={\sin }^{-1}\left(2x\sqrt{1-x^2}\right)$, $\frac{-1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}$, then $\frac{dy}{dx}$is equal to

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

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

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

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

Q. If $y=\frac{1}{{\sin }^{6}x+{\cos }^{6}x},$ then $y$ can be

1. $\frac{3}{2}$
2. $2$
3. $4$
4. $5$

Q.

In a $△ABC,\angle A={55}^o,\angle B={15}^o,\angle C={110}^{o}then{c}^2-a^2$ is equal to

1. ab

2. 2ab

3. -ab

4. None of these