Trigonometry & Inverse Trigonometry
Max-Min of Trig Function — α+2β
nta_pyq_2026_jan
Grade 11
Question:
Let $\alpha$ and $\beta$ respectively be the maximum and minimum values of the function $f(\theta)=4\left(\sin^4\!\left(\dfrac{7\pi}{2}-\theta\right)+\sin^4(11\pi+\theta)\right)-2\left(\sin^6\!\left(\dfrac{3\pi}{2}-\theta\right)+\sin^6(9\pi-\theta)\right)$, $\theta\in\mathbb{R}$. Then $\alpha+2\beta$ is equal to:
Step-by-Step Solution
Key Concept: Simplify: $\sin^4(7\pi/2-\theta)=\cos^4\theta$, $\sin^4(11\pi+\theta)=\sin^4\theta$ (since $\sin(11\pi+\theta)=-\sin\theta$). Similarly for sin⁶ terms. $f(\theta)=4(\cos^4\theta+\sin^4\theta)-2(\cos^6\theta+\sin^6\theta)$.
$\alpha=2$, $\beta=3/2$. $\alpha+2\beta=5$.
Correct Answer: 3