2π - ( sin ⁻¹4/5 + sin ⁻¹5/13 + sin ⁻¹16/65 ) is equal to :

Solution

Maths sir · 4 steps

Combine the first two angles into one sine, and the third one turns out to be its complement.

1. Get the cosines. For α = sin⁻¹ 4/5, cos α = 3/5. For β = sin⁻¹ 5/13, cos β = 12/13. Both angles lie in (0, π/2), so both cosines are positive.

   sin(α+β) = 4/5·12/13 + 3/5·5/13 = (48+15)/65 = 63/65

2. Combine the first two. Since (4/5)² + (5/13)² < 1, the sum α + β stays below π/2. So sin⁻¹ 4/5 + sin⁻¹ 5/13 = sin⁻¹ 63/65.
3. Spot the complement. 16/65 and 63/65 satisfy 16² + 63² = 65². So sin⁻¹ 16/65 = cos⁻¹ 63/65, and sin⁻¹ 63/65 + cos⁻¹ 63/65 = π/2.
4. So. The bracket equals π/2, and 2π - π/2 = 3π/2, which is option (c).

The idea to keep. When inverse sines have Pythagorean-triple values, first merge two of them with the sin(A+B) formula. Then check whether the result and the leftover angle add up to π/2.

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