2 tan - 1 a - b a + b 1 2 tan θ 2 equals

cos - 1 a cos θ + b a + b cos θ

cos - 1 a + b cos θ a cos θ + b

The sum tan - 1 1 1 + x + x 2 + tan - 1 1 3 + 3 x + x 2 + tan - 1 1 7 + 5 x + x 2 + tan - 1 1 13 + 7 x + x 2 + … . of first 100 terms of the series is

tan - 1 2 x - 100 1 - x 2 - 100 x

tan - 1 2 x + 100 1 + x 2 - 100 x

Prove that cos - 1 x = 2 cos - 1 1 + x 2

To prove: cos - 1 x = 2 cos - 1 1 + x 2 Let x = cos 2 θ cos - 1 cos 2 θ = 2 cos - 1 1 + cos 2 θ 2 = 2 cos - 1 2 cos 2 θ 2 = 2 cos - 1 cos θ = 2 θ

Show that sin - 1 3 5 - sin - 1 8 17 = cos - 1 84 85

Let sin − 1 3 5 = x and sin − 1 8 17 = y Therefore sin x = 3 5 and siny = 8 17 Now cos x = 1 − sin 2 x = 1 − 9 25 = 4 5 and cos y = 1 − sin 2 y = 1 − 64 289 = 15 17 We have cos ( x - y ) = cosxcosy + sinxsiny = 4 5 × 15 17 + 3 5 × 8 17 = 84 85 Therefore x − y = cos − 1 84 85 Hence sin − 1 3 5 − sin − 1 8 17 = cos − 1 84 85

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 22 - Inverse Trigonometric Functions>Assignment>Q 60

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Earn 100

The number of positive integers in the domain of $f (x) = \cos^{- 1} ([e^{x}] - 1) + \sin^{- 1} ([e^{x}])$ , where $[\cdot]$ represents greatest integer function, is equal to

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Important Questions on Inverse Trigonometric Functions

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 22 - Inverse Trigonometric Functions>Assignment>Q 1

If

|\cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})| < \frac{π}{3}

, then

x

belongs to the interval

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 22 - Inverse Trigonometric Functions>Assignment>Q 2

The set of values of

x

satisfying

|\sin^{- 1} x| \leq |\cos^{- 1} x|

is

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Gamma Question Bank for Engineering Mathematics>Chapter 22 - Inverse Trigonometric Functions>Assignment>Q 3

The solution of the equation

\cos^{- 1} (x \sqrt{3}) + \cos^{- 1} x = \frac{π}{2}

is given by

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Gamma Question Bank for Engineering Mathematics>Chapter 22 - Inverse Trigonometric Functions>Assignment>Q 4

Minimum value of

n \in N

such that

\cot^{- 1} (\frac{n^{2} - 10 n + 21.6}{π}) > \frac{π}{6}

is

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Gamma Question Bank for Engineering Mathematics>Chapter 22 - Inverse Trigonometric Functions>Assignment>Q 5

If

a, b, c

are real positive numbers and

θ = \tan^{- 1} {[\frac{a (a + b + c)}{b c}]}^{\frac{1}{2}} + \tan^{- 1} {[\frac{b (a + b + c)}{c a}]}^{\frac{1}{2}} + \tan^{- 1} {[\frac{c (a + b + c)}{a b}]}^{\frac{1}{2}}

, then

\tan θ

equals

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Gamma Question Bank for Engineering Mathematics>Chapter 22 - Inverse Trigonometric Functions>Assignment>Q 6

Let

\cos^{- 1} \frac{x}{2} + \cos^{- 1} \frac{y}{3} = θ

and denote by

f (x, y, θ) = 0

the rational integral expression in

x

and

y

. Then for

θ = \frac{π}{2}

the locus represented by

f (x, y, \frac{π}{2}) = 0

is

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Gamma Question Bank for Engineering Mathematics>Chapter 22 - Inverse Trigonometric Functions>Assignment>Q 7

2 \tan^{- 1} [{(\frac{a - b}{a + b})}^{\frac{1}{2}} \tan (\frac{θ}{2})]

equals

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 22 - Inverse Trigonometric Functions>Assignment>Q 8

The sum $\tan^{- 1} \frac{1}{1 + x + x^{2}} + \tan^{- 1} (\frac{1}{3 + 3 x + x^{2}}) + \tan^{- 1} (\frac{1}{7 + 5 x + x^{2}}) + \tan^{- 1} \frac{1}{13 + 7 x + x^{2}} + \dots .$