Find the principal value of tan - 1 ( - 3 ) .

Let y = tan - 1 ( - 3 ) ⇒ tan y = - 3 ⇒ tan y = - tan π 3 ⇒ tan y = tan - π 3 . . . . i We know that the general solution of tan θ = tan α is θ = n π + α , where n ∈ Z . Thus, the general solution of equation i is y = n π - π 3 , n ∈ Z ⇒ y = . . . . , - 4 π 3 , - π 3 , 2 π 3 , . . . . Now, the range of tan - 1 x is - π 2 , π 2 . Since - π 3 ∈ - π 2 , π 2 Hence, the principle value of the given trigonometric function is - π 3 .

Show that sin - 1 5 13 + cos - 1 3 5 = tan - 1 63 16

We have to show: sin - 1 5 13 + cos - 1 3 5 = tan - 1 63 16 Let, sin - 1 5 13 = x ⇒ sin x = 5 13 And cos 2 x = 1 - sin 2 x = 1 - 25 169 = 144 169 ⇒ cos x = 144 169 = 12 13 ∴ tan x = sin x cos x = 5 13 12 13 = 5 12 Again, let cos - 1 3 5 = y ⇒ cos y = 3 5 ∴ sin y = 1 - cos 2 y = 1 - 3 5 2 = 1 - 9 25 ⇒ sin y = 16 25 = 4 5 ⇒ tan y = sin y cos y = 4 5 3 5 = 4 3 We know that, tan ( x + y ) = tan x + tan y 1 - tan x tan y ∴ tan ( x + y ) = 5 12 + 4 3 1 - 5 12 4 3 ⇒ tan ( x + y ) = 5 + 16 12 36 - 20 36 ⇒ tan ( x + y ) = 21 × 3 16 ⇒ tan ( x + y ) = 63 16 ⇒ x + y = tan - 1 63 16 ⇒ sin - 1 5 13 + cos - 1 3 5 = tan - 1 63 16 Hence, proved.

MEDIUM

Earn 100

Write down the general value of ${cosec}^{- 1} (- 2)$

Important Questions on Inverse Trigonometric Functions

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

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

Question Image

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

The domain of the function defined by

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

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

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

then the smallest interval in which

θ

lies is

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

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

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

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

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

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

then value of

x

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

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	$\{\sec^{- 1} 1, \sec^{- 1} 2\}$

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

Write down the range of $f (x) = \cot^{- 1} x$ .

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

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

then the value of

x^{100} + y^{100} + z^{100} - \frac{9}{x^{101} + y^{101} + z^{101}}

is equal to

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

\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>Inverse Trigonometric Functions and their Graphs

\cos [\cot^{- 1} (- \sqrt{3}) + \frac{π}{6}] =

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

The value of

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

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

The domain of the function,

f (x) = \sin^{- 1} (\frac{3 x^{2} + x - 1}{(x - 1)^{2}}) + \cos^{- 1} (\frac{x - 1}{x + 1})

is:

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

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>Inverse Trigonometric Functions and their Graphs

The principal value of

\tan^{- 1} (\tan \frac{3 π}{5})

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

Find the principal value of

\tan^{- 1} (- \sqrt{3})

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

Show that

\sin^{- 1} (\frac{5}{13}) + \cos^{- 1} (\frac{3}{5}) = \tan^{- 1} (\frac{63}{16})

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

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

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

α = 3 \sin^{- 1} (\frac{6}{11}) & β = 3 \cos^{- 1} (\frac{4}{9}),

where the inverse trigonometric functions take only the principal values, then the correct options(s) is(are)

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Functions and their Graphs

α

and

β

are roots of the equation

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

then the value of

|\tan^{- 1} α - \tan^{- 1} β|

Write down the general value of cosec-1-2

Important Questions on Inverse Trigonometric Functions

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Write down the general value of ${cosec}^{- 1} (- 2)$