Considering only the principal values of the inverse trigonometric functions, the value of

$\frac{3}{2} \cos^{- 1} \sqrt{\frac{2}{2 + π^{2}}} + \frac{1}{4} \sin^{- 1} (\frac{2 \sqrt{2} π}{2 + π^{2}}) + \tan^{- 1} (\frac{\sqrt{2}}{π})$ is ________. [2022]

Question

Considering only the principal values of the inverse trigonometric functions, the value of

$\frac{3}{2} \cos^{- 1} \sqrt{\frac{2}{2 + π^{2}}} + \frac{1}{4} \sin^{- 1} (\frac{2 \sqrt{2} π}{2 + π^{2}}) + \tan^{- 1} (\frac{\sqrt{2}}{π})$ is ________. [2022]

Smart Achievers · Accepted Answer

(2.36)

$[Let, \cos^{- 1} \sqrt{\frac{2}{2 + π^{2}}} = t = \tan^{- 1} (\frac{π}{\sqrt{2}})]$

${Similarly for \sin^{- 1} (\frac{2 \sqrt{2} π}{2 + π^{2}})}$

$Now, we have$

$\frac{3}{2} \tan^{- 1} (\frac{π}{\sqrt{2}}) + \frac{1}{4} \tan^{- 1} (\frac{2 \sqrt{2} π}{π^{2} - 2}) + \tan^{- 1} (\frac{\sqrt{2}}{π})$

$= \frac{π}{2} + \frac{1}{2} \tan^{- 1} (\frac{π}{\sqrt{2}}) - \frac{1}{4} \tan^{- 1} (\frac{2 \sqrt{2} π}{2 - π^{2}})$

$= \frac{π}{2} + \frac{1}{2} \tan^{- 1} (\frac{π}{\sqrt{2}}) - \frac{1}{4} \tan^{- 1} (\frac{2 (\frac{π}{\sqrt{2}})}{1 - {(\frac{π}{\sqrt{2}})}^{2}})$

$= \frac{π}{2} + \frac{1}{2} \tan^{- 1} (\frac{π}{\sqrt{2}}) - \frac{1}{4} (- π + 2 \tan^{- 1} (\frac{π}{\sqrt{2}}))$

$= \frac{π}{2} + \frac{π}{4} = \frac{3 π}{4} = 2.36$

Solution

Q.
Considering only the principal values of the inverse trigonometric functions, the value of

$\frac{3}{2} \cos^{- 1} \sqrt{\frac{2}{2 + π^{2}}} + \frac{1}{4} \sin^{- 1} (\frac{2 \sqrt{2} π}{2 + π^{2}}) + \tan^{- 1} (\frac{\sqrt{2}}{π})$ is ________. [2022]

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Solution

Q. Considering only the principal values of the inverse trigonometric functions, the value of 32cos-122+π2+14sin-1(22π2+π2)+tan-1(2π) is ________. [2022]

Ans. (2.36) [Let, cos-122+π2=t=tan-1(π2)] {Similarly for sin-1(22π2+π2)} Now, we have 32tan-1(π2)+14tan-1(22ππ2-2)+tan-1(2π) =π2+12tan-1(π2)-14tan-1(22π2-π2) =π2+12tan-1(π2)-14tan-1(2(π2)1-(π2)2) =π2+12tan-1(π2)-14(-π+2tan-1(π2)) =π2+π4=3π4=2.36

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Q.
Considering only the principal values of the inverse trigonometric functions, the value of

$\frac{3}{2} \cos^{- 1} \sqrt{\frac{2}{2 + π^{2}}} + \frac{1}{4} \sin^{- 1} (\frac{2 \sqrt{2} π}{2 + π^{2}}) + \tan^{- 1} (\frac{\sqrt{2}}{π})$ is ________. [2022]