JEE Main · 2021 · 01 Sep 2021 · Shift 2 · Inverse Trigonometric Functions

cos^{- 1} (cos (- 5)) + sin^{- 1} (sin (6)) - tan^{- 1} (tan (12))

is equal to:
(The inverse trigonometric functions take the principal values)

Answer: (C)

Solution

Using principal value ranges:

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

. Since

5 \in (π, 2 π)

, we have

cos^{- 1} (cos 5) = 2 π - 5

sin^{- 1} (sin 6)

. Since

6 \in (3 π /2, 2 π)

, we have

sin^{- 1} (sin 6) = 6 - 2 π

tan^{- 1} (tan 12)

. Since

12 \in (3 π, 4 π)

, we have

tan^{- 1} (tan 12) = 12 - 4 π

.

Therefore:

(2 π - 5) + (6 - 2 π) - (12 - 4 π) = 2 π - 5 + 6 - 2 π - 12 + 4 π = 4 π - 11

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