If sin ⁻¹α/17 + cos ⁻¹4/5 - tan ⁻¹77/36 = 0,0 < α < 13, then sin ⁻¹(sin α ) + cos ⁻¹(cos α ) is equal to :

Solution

Maths sir · 5 steps

Find α = 8 first, then pull 8 radians back into each principal range; never cancel the inverse blindly.

1. Rewrite in tan. cos⁻¹ 4/5 is an angle with cos = 4/5 and sin = 3/5, so it equals tan⁻¹ 3/4. The equation becomes sin⁻¹ α/17 = tan⁻¹ 77/36 - tan⁻¹ 3/4.

   tan⁻¹ 77/36 - tan⁻¹ 3/4 = tan⁻¹((77/36-3/4)/(1+77/36×3/4)) = tan⁻¹((25/18)/(125/48)) = tan⁻¹ 8/15

2. Find alpha. An angle with tan = 8/15 has hypotenuse 17, so its sine is 8/17. Then α/17 = 8/17, giving α = 8, which lies in (0, 13).
3. Locate 8 radians. 8 - 2π ≈ 1.717, which lies between π/2 ≈ 1.571 and π. So cos⁻¹(cos 8) = 8 - 2π, since that value is already in [0, π].
4. The sine part. 8 - 2π is past π/2, so use sin(π - θ) = sin θ. The angle π - (8 - 2π) = 3π - 8 ≈ 1.425 lies in [-π/2, π/2], so sin⁻¹(sin 8) = 3π - 8.
5. So. The sum is (3π - 8) + (8 - 2π) = π, which is option (b).

The idea to keep. Never cancel sin⁻¹(sin x) blindly. Shift x by 2π and reflect until it lands in [-π/2, π/2] for sine or [0, π] for cosine.

Answer from the NTA final answer key. Solution by PrepWiser.