Trigonometry & Inverse Trigonometry
Trigonometric Ratios & Identities
nta_pyq_2025_jan
Grade 11
Question:
If \pi \le x \le 3\pi , then cos -1 ( 12 cos x + 5 sin x) is equal to 2 4 13 13
x - tan -1 4 3
x + tan -1 4 5
x - tan -1 12 5
x + tan -1 12 5
Step-by-Step Solution
Key Concept: Apply the core result for trigonometric identities and simplify using the given constraints.
12 cos x + 5 sin x 13 13 (3) 5 \pi Let tan \alpha = , \alpha \in (0, ) 12 2 5 12 \Rightarrow sin \alpha = , cos \alpha = 13 13 12 5 \Rightarrow cos x + sin x = cos \alpha cos x + sin \alpha sin x 13 13 = cos(x - \alpha) -1 \Rightarrow cos [cos(x - \alpha)] = x - \alpha 5 -1 = x - tan ( ) 12
Correct Answer: 3