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The principal value of $\cos^{- 1} (\frac{- 1}{2})$ is

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Important Questions on Inverse Trigonometric Functions

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

The value of

\cos (\sin^{- 1} \frac{π}{3} + \cos^{- 1} \frac{π}{3})

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

\sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = \frac{3 π}{2}

and

f (1) = 2, f (p + q) = f (p) \cdot f (q), \forall p, q \in R

, then

x^{f (1)} + y^{f (2)} + z^{f (3)} - \frac{(x + y + z)}{x^{f (1)} + y^{f (2)} + z^{f (3)}}

is equal to

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

The domain of the function defined by

f (x) = \cos^{- 1} \sqrt{x - 1}

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

Using the information from the figure, $\cos^{- 1} x$ is equal to

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Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

α

and

β

are roots of the equation

x^{2} + 5 |x| - 6 = 0

then the value of

|\tan^{- 1} α - \tan^{- 1} β|

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

θ = \sin^{- 1} x + \cos^{- 1} x - \tan^{- 1} x, x \geq 0

then the smallest interval in which

θ

lies is

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

Let $Z$ denote the set of integers. Then match the items in list- $I$ , with those of the items in List- $II$

	List- $I$		List- $II$
$(A)$	$\sin^{- 1} (\frac{2 \sqrt{2}}{3}) + \sin^{- 1} \frac{1}{3}$	$(I)$	$k π \pm {(- 1)}^{k} \frac{π}{6}, k \in Z$
$(B)$	$\sin^{- 1} (\frac{{(- 1)}^{n}}{2}), n \in Z$	$(II)$	$k π \pm 1, k \in Z$
$(C)$	$\tan^{- 1} (\sec \frac{π}{4} + \tan \frac{π}{4})$	$(III)$	$\frac{3}{2}$
$(D)$	$\sin^{- 1} \|\sin x\| = \sqrt{\sin^{- 1} \|\sin x\|} \Rightarrow x \in$	$(IV)$	$\frac{3 π}{8}$
		$(V)$	$\frac{π}{2}$

The correct match is

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

\tan^{- 1} [\frac{1}{\sqrt{3}} \sin \frac{5 π}{2}] \sin^{- 1} [\cos (\sin^{- 1} \frac{\sqrt{3}}{2})] =

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

The value of

\cos^{- 1} (\cot (\frac{π}{2})) + \cos^{- 1} (\sin (\frac{2 π}{3}))

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

The domain of the function

f (x) = \sin^{- 1} (\frac{|x| + 5}{x^{2} + 1})

(- \infty, - a] \cup [a, \infty)

, then

a

is equal to

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

The number of solutions of the equation

\sin^{- 1} [x^{2} + \frac{1}{3}] + \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 :

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

The principal value of

s i n^{- 1} (\sin \frac{5 π}{3})

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

\cos (2 \tan^{- 1} x) = \frac{1}{2},

then value of

x

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Basics of Inverse Trigonometric Functions

Match the items of List-I with those of the items of List-II

	List- $I$		List- $II$
A	Range of $\sec^{- 1} [1 + \cos^{2} x]$ , $[\cdot]$ denote greatest integer function	I	odd function
B	Domain of $f (x)$ where $f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}$	II	$\{0, \frac{1}{2}\}$
C	$f (x + y) = f (x) + f (y); f (1) = 5$	III	$\{\sec^{- 1} 5, \sec^{- 1} 4\}$
D	$\sin^{- 1} x - \cos^{- 1} x + \sin^{- 1} (1 - x) = 0 \Rightarrow x \in$	IV	$R$
		V	$\{\sec^{- 1} 1, \sec^{- 1} 2\}$