sin ¹( sin 2π/3 ) + cos ⁻¹( cos 7π/6 ) + tan ⁻¹( tan 3π/4 ) is equal to :

Solution

Maths sir · 2 steps

The inverse undoes the function only inside its principal range, so pull each angle into that range first.

1. The rule. sin⁻¹(sin θ) = θ only when θ lies in [-π/2, π/2]. The same holds for cos⁻¹ on [0, π] and tan⁻¹ on (-π/2, π/2). Otherwise, find the angle in that range that has the same sine, cosine or tangent.

   Term	Principal range	Value of the inner function	Angle in range
   sin⁻¹(sin 2π/3)	[-π/2, π/2]	sin 2π/3 = √3/2	π - 2π/3 = π/3
   cos⁻¹(cos 7π/6)	[0, π]	cos 7π/6 = -√3/2	2π - 7π/6 = 5π/6
   tan⁻¹(tan 3π/4)	(-π/2, π/2)	tan 3π/4 = -1	3π/4 - π = -π/4

   π/3 + 5π/6 - π/4 = (4π + 10π - 3π)/12

2. So. The sum is 11π/12, which is option (a).

The idea to keep. Before cancelling an inverse with its function, check the principal range. If the angle lies outside it, shift or reflect it into the range while keeping the same sin, cos or tan value.

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