If π/2 ≤ x ≤ (3 π)/4, then cos ⁻¹(12/13 cos x+5/13 sin x) is equal to

Solution

Maths sir · 4 steps

Spot 12/13 and 5/13 as cos and sin of one angle, and the bracket becomes cos(x − α).

1. Name the angle. Since 12² + 5² = 13², take an angle α with cos α = 12/13 and sin α = 5/13. Then α = tan⁻¹ 5/12, which is about 0.39 rad (roughly 22.6°).
2. Compound angle. The bracket becomes cos x cos α + sin x sin α, which is the expansion of cos(x - α).

   cos⁻¹(12/13 cos x + 5/13 sin x) = cos⁻¹(cos(x - α))

3. Check the range. cos⁻¹(cos θ) = θ only when θ lies in [0, π]. For π/2 ≤ x ≤ 3π/4, x - α runs from about 1.18 to about 1.96, well inside [0, π].
4. So. The expression equals x - α = x - tan⁻¹ 5/12, which is option (d).

The idea to keep. When you see a cos x + b sin x with a² + b² = 1, write it as cos(x - α). Then check that x - α lies in [0, π] before cancelling cos⁻¹ with cos.

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