Inverse Trigonometry JEE Main Previous Year Trigonometry Questions and Solutions - Solution of Triangles

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Solution

Q.
If $a = \sin^{- 1} (\sin (5)) and b = \cos^{- 1} (\cos (5)),$ then $a^{2} + b^{2}$ is equal to [2024]

1 $25$

2 $8 π^{2} - 40 π + 50$

3 $4 π^{2} - 20 π + 50$

4 $4 π^{2} + 25$

Ans.
(2)

We have, $a = \sin^{- 1} (\sin (5)) and b = \cos^{- 1} (\cos (5))$

$\Rightarrow a = - 2 π + 5$ $(∵ \sin^{- 1} (\sin x) = - 2 π + x, if \frac{3 π}{2} \leq x \leq \frac{5 π}{2})$

$b = 2 π - 5,$ $(∵ \cos^{- 1} (\cos x) = 2 π - x, π \leq x \leq 2 π)$

$∴$ $a^{2} + b^{2} = {(- 2 π + 5)}^{2} + {(2 π - 5)}^{2}$

$= 4 π^{2} + 25 - 20 π + 4 π^{2} + 25 - 20 π = 8 π^{2} - 40 π + 50$

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