Trigonometry & Inverse Trigonometry
Number of Solutions — Principal Value Range
nta_pyq_2024_apr
Grade 12
Question:
Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2\sin^{-1}x+3\cos^{-1}x=\dfrac{2\pi}{5}$, is
Step-by-Step Solution
Key Concept: Use $\sin^{-1}x+\cos^{-1}x=\pi/2$ so $2\sin^{-1}x+3\cos^{-1}x=2\sin^{-1}x+3(\pi/2-\sin^{-1}x)=3\pi/2-\sin^{-1}x$. Equivalently, $\pi+\cos^{-1}x=2\pi/5$.
$\cos^{-1}x=-3\pi/5$ is outside $[0,\pi]$. Number of solutions $=0$.
Correct Answer: 0