Evaluate cos-1x-cos-1x-2-1-2-3-3x2-x- - 1-2-1 -

The correct option is D +π/3 Let nbsp; z = nbsp; cos^ 1x+cos^ 1{x/2+1/2√(3 3x^2)} and nbsp; t =cos^ 1{x/2+1/2√(3 3x^2)} Let x = cosθ⇒ ...

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Byju's Answer

Standard XII

Mathematics

Integration by Parts

Evaluate co...

Question

Evaluate ${cos}^{- 1} x + {cos}^{- 1} {\frac{x}{2} + \frac{1}{2} \sqrt{3 - 3 x^{2}}}, x ϵ [\frac{1}{2}, 1]$

- \frac{π}{6}

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+ \frac{π}{6}

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- \frac{π}{3}

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+ \frac{π}{3}

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Solution

The correct option is D $+ \frac{π}{3}$
Let $z = {cos}^{- 1} x + {cos}^{- 1} {\frac{x}{2} + \frac{1}{2} \sqrt{3 - 3 x^{2}}}$
and $t = {cos}^{- 1} {\frac{x}{2} + \frac{1}{2} \sqrt{3 - 3 x^{2}}}$
Let $x = cos θ \Rightarrow t = {cos}^{- 1} (\frac{cos θ}{2} + \frac{\sqrt{3} sin θ}{2})$
$= {cos}^{- 1} cos (\frac{π}{3} - θ) = \frac{π}{3} - θ = \frac{π}{3} - {cos}^{- 1} x$
$∴ z = \frac{π}{3}$

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Similar questions

Q. Prove that :

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

Inverse circular functions,Principal values of

{s i n}^{- 1} x, {c o s}^{- 1} x, {t a n}^{- 1} x

{t a n}^{- 1} x + {t a n}^{- 1} y = {t a n}^{- 1} \frac{x + y}{1 - x y}

x y < 1

π + {t a n}^{- 1} \frac{x + y}{1 - x y}

x y > 1

Evaluate

(a)

{c o s}^{- 1} x + {c o s}^{- 1} [\frac{x}{2} + \frac{\sqrt{(3 - 3 x^{2})}}{2}]

(\frac{1}{2} \leq x \leq 1)

(b)

c o s (2 {c o s}^{- 1} x + {s i n}^{- 1} x)

x = 1 / 5

where

0 \leq {c o s}^{- 1} x \leq π

and

- π / 2 \leq {s i n}^{- 1} x \leq π / 2

Inverse circular functions,Principal values of ${sin}^{- 1} x, {c o s}^{- 1} x, {tan}^{- 1} x$ .
${tan}^{- 1} x + {tan}^{- 1} y = {tan}^{- 1} \frac{x + y}{1 - x y}$ , $x y < 1$
$π + {tan}^{- 1} \frac{x + y}{1 - x y}$ , $x y > 1$ .
Evaluate the following :
(a) $sin [\frac{π}{3} - {sin}^{- 1} (- \frac{1}{2})]$
(b) $sin [\frac{π}{2} - {sin}^{- 1} (- \frac{\sqrt{3}}{2})]$

Q. Number of integral solution of the equation

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

is -

Q. Prove that if

\frac{1}{2} \leq x \leq 1

then,

c o s^{- 1} x + c o s^{- 1} [\frac{x}{2} + \frac{\sqrt{3 - 3 x^{2}}}{2}] = \frac{π}{3} .

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Evaluate cos−1x+cos−1{x2+12√3−3x2},xϵ[12,1]

The correct option is D +π3Let z=cos−1x+cos−1{x2+12√3−3x2}and t=cos−1{x2+12√3−3x2}Let x=cosθ⇒t=cos−1(cosθ2+√3sinθ2)=cos−1cos(π3−θ)=π3−θ=π3−cos−1x∴z=π3

Evaluate ${cos}^{- 1} x + {cos}^{- 1} {\frac{x}{2} + \frac{1}{2} \sqrt{3 - 3 x^{2}}}, x ϵ [\frac{1}{2}, 1]$