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

JEE Main · 2024 · 31 Jan 2024 · Shift 2 · Inverse Trigonometric Functions

a = sin^{- 1} (sin (5))

and

b = cos^{- 1} (cos (5))

, then

a^{2} + b^{2}

is equal to

Answer: (B)

5

radians lies between

\frac{3 π}{2} \approx 4.712

and

2 π \approx 6.283

.

For

x \in [\frac{3 π}{2}, 2 π]

sin^{- 1} (sin x) = x - 2 π

(this brings the value into

[- \frac{π}{2}, \frac{π}{2}]

), so

a = 5 - 2 π

.

For

x \in [π, 2 π]

cos^{- 1} (cos x) = 2 π - x

(bringing it into

[0, π]

); since

5 \in [π, 2 π]

b = 2 π - 5

.

Note

b = - a

, so

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

(5 - 2 π)^{2} = 25 - 20 π + 4 π^{2}

a^{2} + b^{2} = 2 (25 - 20 π + 4 π^{2}) = 8 π^{2} - 40 π + 50

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