Domain and Range of Basic Inverse Trigonometric Functions

Q. Range of f(x) =
${\sin }^{-1}x+{\tan }^{-1}x+{\sec }^{-1}x\mathrm{is}$

1. $\left(\frac{\pi }{4},\frac{3\pi }{4}\right)$
2. $\left(\frac{\pi }{4},\frac{3\pi }{4}\right]$
3. $\{\frac{\pi }{4},\frac{3\pi }{4}\}$
4. none of these

Q. Consider the function $f\left(x\right)={\cos }^{-1}\left(\right[2^x\left]\right)+{\sin }^{-1}\left(\right[2^x-1\left]\right)$ (where $[.]$ denotes greatest integer function), then

1. Domain of $f\left(x\right)$ is $x\in (-\infty ,0]$
2. Range of $f\left(x\right)$ is singleton set
3. $f\left(x\right)$ is an even function
4. $f\left(x\right)$ is an odd function

Q. Q.34 If f(x) and g(x) are periodic functions with the same fundamental period where f(x) = sinαx + cosαx and g(x) = |sinx| + |cosx|, then α is equal to

Q. Express each of the following as the product of sines and cosines:
(i) sin 12x + sin 4x
(ii) sin 5x − sin x
(iii) cos 12x + cos 8x
(iv) cos 12x − cos 4x
(v) sin 2x + cos 4x

Q. If $\left[si{n}^{-1}x\right]>\left[co{s}^{-1}x\right]$ where [.] denotes the greatest integer function, then the set of values of x is:

1. [cos 1, 1]
2. [cos 1, sin 1]
3. [sin 1, 1]
4. None of these

Q. The trigonometric equation $si{n}^{-1}x=2si{n}^{-1}$ a has a solution for:

1. all real values of a
2. $|a|<\frac{1}{2}$
3. $|a|\le \frac{1}{\sqrt{2}}$
4. $\frac{1}{2}<|a|<\frac{1}{\sqrt{2}}$

Q. If $si{n}^{-1}x+si{n}^{-1}y+si{n}^{-1}z=\frac{3\pi }{2}$, then xy +yx -xz =

1. \N
2. 1
3. 1 or -1
4. None

Q.

Which of the following statements is/are correct?

1. cos2A = $\frac{1+ta{n}^{2}A}{1-ta{n}^{2}A}$

2. $cose{c}^{2}A$ = 1 + $co{t}^2\theta$

3. $se{c}^2\theta$ = 1 + $co{s}^2\theta$

1. Only 1

2. Only 2

3. Only 1 & 2

4. Only 2 & 3

Q.

How many of the following relations are correct?

(1)1. Sin(A+B) = sinAcosB + cosAsinB

(2)2. cos(A-B) = CosAcosB - sinAsinB

(1)3. Tan (A-B) = $\frac{tanA-tanB}{1+tanAtanB}$

(2)4. Sin(A+B) (sin(A-B) = $co{s}^{2}A-co{s}^{2}B$

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Q. all trigonometric formulaes

Q. If $f\left(x\right)=({\sin }^{-1}x{)}^2+({\cos }^{-1}x{)}^2$, then

1. $f\left(x\right)$ has the least value of $\frac{{\pi }^2}{8}$
2. $f\left(x\right)$ has the greatest value of $\frac{5{\pi }^2}{8}$
3. $f\left(x\right)$ has the least value of $\frac{{\pi }^2}{16}$
4. $f\left(x\right)$ has the greatest value of $\frac{5{\pi }^2}{4}$

Q. If $si{n}^{-1}x+si{n}^{-1}y+si{n}^{-1}z=\frac{3\pi }{2}$
then $\frac{{\sum }_{r=1}^2(x^{100r}+y^{103r})}{\sum x^{201}{y}^{201}}$ =

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

Q.

If $z+\frac{1}{z}=\sqrt{3}$ then ${\sum }_{r=1}^5{(z^r+\frac{1}{z^r})}^2$ =

1. 10

2. 8

3. 15

4. 12

Q. Find the general solution for the trigonometric equation $\cos 2x+\cos x=0.$

1. $x\in \{2n\pi \pm \frac{\pi }{3}\}\cup \{2n\pi \pm \frac{\pi }{2}\},n\in Z$
2. $x\in \{2n\pi \pm \frac{\pi }{3}\}\cup \{2n\pi \pm \pi \},n\in Z$
3. $x\in \{2n\pi +\frac{\pi }{3}\}\cup \{2n\pi +\pi \},n\in Z$
4. None of the above.

Q. If $y=si{n}^{-1}x+ta{n}^{-1}x+se{c}^{-1}x$, then set of values of y is

1. $-\frac{\pi }{2},\frac{3\pi }{4}$
   
2. $\frac{-3\pi }{4},\frac{3\pi }{4}$
   
3. $\frac{\pi }{4},\frac{3\pi }{4}$
   
4. $\frac{-\pi }{4},\frac{\pi }{4}$

Q. If $co{t}^{-1}\frac{n}{\pi }>\frac{\pi }{6},nϵN$, then the maximum value of n =

1. 6
2. 5
3. 4
4. None

Q. The domain of the function ${\cos }^{-1}(3x-2)$ is

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

Q. If $xϵ[-1,1],thenrangeofta{n}^{-1}(-x)$ is

1. $[\frac{3\pi }{4},\frac{7\pi }{4}]$
2. $[\frac{3\pi }{4},\frac{5\pi }{4}]$
3. $[\pi ,0]$
4. $[\frac{-\pi }{4},\frac{\pi }{4}]$

Q. The range of $f\left(x\right)={\cos }^{-1}x+2{\tan }^{-1}x+3{\mathrm{cosec}}^{-1}x$ is

1. $[-\pi ,2\pi ]$
2. $\varphi$
3. $(-\pi ,2\pi )$
4. $\{-\pi ,2\pi \}$

Q. If $2^{\frac{2\pi }{{\sin }^{-1}x}}-2(a+2)2^{\frac{\pi }{{\sin }^{-1}x}}+8a<0$ for atleast one real $x$, then

1. $\frac{1}{8}\le a<2$
2. $a<2$
3. $a\in R-\left\{2\right\}$
4. $a\in [0,\frac{1}{8})\cup (2,\infty )$

Q. The number of solutions of the equation $ta{n}^{-1}\left(\frac{x}{1-x^2}\right)+ta{n}^{-1}\left(\frac{1}{x^3}\right)=\frac{3\pi }{4}$ belonging to the interval (0, 1) is

1. \N
2. 1
3. 2
4. 3