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Inverse Trigonometric Functions Important questions




Multiple Choice Questions

Question 1
What is the range of the inverse sine function $sin^{-1} x$
(a) $-\pi \leq sin^{-1} x \leq \pi$
(b) $-\frac {\pi}{2} \leq sin^{-1} x \leq \frac {\pi}{2} $
(c) $0 \leq sin^{-1} x \leq \leq \pi $
(d) $-\pi \ge sin^{-1} x < \pi$

Question 2
Which of the following statements is true about the arc cosine function?
(a) $cos^{-1}(-x) = -cos^{-1}x $
(b) $cos^{-1}(-x) = \pi - cos^{-1}x $
(c) $cos^{-1}(-x) = cos^{-1}x $
(d) $cos^{-1}(-x) = \pi + cos^{-1}x$

Question 3
If x and y are positive acute angles such that cos(x) = sin(y), then which of the following is true?
(a) $x + y = \pi/2$
(b) x = y
(c) $x = \pi/2 - y$
(d) $x + y = \pi$

Question 4
What is the value of $sin^{-1}(-1/2)$?
(a) -π/6
(b) π/6
(c) -π/3
(d) π/3

Question 5
If $tan^{-1}(x) = \theta$, then which of the following is true?
(a) $sin(\theta) = \frac {x}{\sqrt{(1 + x^2)}}$
(b) $sin(\theta) = \frac {x}{\sqrt {(1 - x^2)}}$
(c) $cos(\theta) = \frac {x}{\sqrt {(1 + x^2)}}$
(d) $cos(\theta) = \frac {x}{\sqrt {(1 - x^2)}}$

Question 6
The domain of the function $y = sin^{–1} (-x^2)$ is
(a)[0, 1]
(b) (0, 1)
(c) [–1, 1]
(d) $\phi$

Question 7
If $sin^{-1} x + sin^{-1} y = \frac {\pi}{2}$ , then value of $cos^{-1} x + cos^{-1} y$ is
(a) $\frac {\pi}{2}$
(b) $-\frac {\pi}{2}$
(d) 0
(d) $\pi$

Question 8

if $ A= tan^{-1} tan \frac {5\pi}{4}$ and $B= tan^{-1} (-tan \frac {2\pi}{3})$, then
(a) 2A=3B
(b) 4A=3B
(c) A=2B
(d) 6A=B

Question 9
The greatest and least value of the function $f(x) = (sin^{-1}x)^2 + (cos^{-1}x)^2$ are
(a)$\frac{\pi ^2}{4}$, 0
(b) $\frac{5\pi ^2}{4}$, $\frac{\pi ^2}{8}$
(c) $\frac{\pi ^2}{4}$, $\frac{\pi ^2}{8}$
(d) $\frac{3\pi ^2}{4}$, $\frac{\pi ^2}{8}$

Question 10
The value of $sin^{-1} ( cos \frac {33\pi}{5})$ is
(a) $\frac {3\pi}{5}$
(b) $-\frac {\pi}{10}$
(c) $\frac {\pi}{10}$
(d) $\frac {7\pi}{5}$

Fill in the blanks

Question 11
(i) The value of $tan^2(sec^{–1} 2) + cot^2 (cosec^{–1} 3)$ is ________
(ii) The principal value of $tan^{-1} \sqrt 3$ is _____
(iii) The value of $sin (sin^{-1} x + cos^{-1} x)$ , $|x| \leq 1$ is ________
(iv) The value of $cos^{-1} (cos \frac {5\pi}{3}) + sin^{-1} (sin \frac {5\pi}{3})$ is _____

Long answer questions

Question 12
Prove that
$tan^{-1}( \frac { \sqrt {1+ cosx} + \sqrt {1-cos x}}{ \sqrt {1+ cosx} - \sqrt {1-cosx}}) =\frac {\pi}{4} - \frac {x}{2}$
Here $\pi < x < \frac {3\pi}{2}$

Question 13
Prove that
$2sin^{-1} \frac {3}{5} - tan^{-1} \frac {17}{31} = \frac {\pi}{2}$

Question 14
Solve the equation
$cos(tan^{-1} x) = sin (cot^{-1} \frac {3}{4})$

Question 15
Solve the equation
$tan^{-1} (x-1) + tan^{-1} x + tan^{-1} (x+1) =tan^{-1} 3x$

Question 16
Prove that
$tan^{-1} (\frac {6x- 8x^3}{1-12x^2}) - tan^{-1} ( \frac {4x}{1-4x^2})=tan^{-1} 2x$ ; $|2x| < \frac {1}{\sqrt 3}$
Question 17
Solve the equation
$tan^{-1} (\frac {2-x}{2+x}) = \frac {1}{2} tan^{-1} \frac {x}{2}$ , x > 0

Question 18
Solve the equation
$sin^{-1} x + sin^{-}1 (1 - x) = cos^{-1} x$

Question 19
Prove that
$tan^{-1}( \frac { \sqrt {1+ x} - \sqrt {1- x}}{ \sqrt {1+x} + \sqrt {1-x}} =\frac {\pi}{4} -\frac {1}{2} cos^{-1}x$
Here $-\frac {1}{\sqrt 2} \leq x \leq 1$

Question 20
Solve the equation
$tan^{-1} 2x +tan^{-1} 3x = \frac {\pi}{4}$

Question 21
Prove that
$sin ^{-1} \frac {4}{5} +sin ^{-1} \frac {5}{13} + sin ^{-1} \frac {16}{65}= \frac {\pi}{2}$

Question 22
Prove that
$sin ^{-1} \frac {4}{5} + tan ^{-1} \frac {5}{12} + cos ^{-1} \frac {63}{65}= \frac {\pi}{2}$

Question 23
Solve the equation
$cot^{-1}7 + cot^{-1}8 + cot^{-1} 18 = cot^{-1}3$

Question 24
Solve the equation
$tan^{-1} (2 +x ) + tan^{-1} (2 -x )= tan^{-} \frac {2}{3}$ ,$- \sqrt 3 < x < \sqrt 3$

Question 24
Solve the equation
$sin^{-1} (x ) + sin^{-1} (2x )= \frac { \pi}{3}$ , x > 0


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