Trigonometric Equations
Equation Involving sin and cos — Number of Solutions
nta_pyq_2024_jan
Grade 11
Question:
The number of solutions of the equation $4\sin^2x-4\cos^3x+9-4\cos x=0$; $x\in[-2\pi,2\pi]$ is:
Step-by-Step Solution
Key Concept: Rewrite using $\sin^2x=1-\cos^2x$: $4(1-\cos^2x)-4\cos^3x+9-4\cos x=0\Rightarrow4\cos^3x+4\cos^2x+4\cos x-13=0$. LHS is bounded: maximum of $4\cos^3x+4\cos^2x+4\cos x\leq12<13$. No solution.
LHS $\leq12<13$. No solutions.
Correct Answer: 4