Trigonometry
Sum of solutions of trig equation; $\sin^2+\cos^4$ type
Grade Class 12
Question:
Sum of all values of $\theta\in\left(0,\dfrac{\pi}{2}\right)$ satisfying $\sin^22\theta+\cos^42\theta=\dfrac{3}{4}$ is
$\dfrac{\pi}{2}$
$\pi$
$\dfrac{3\pi}{8}$
$\dfrac{5\pi}{4}$
Step-by-Step Solution
Key Concept: Let $u=\cos^22\theta$. Equation: $(1-u)+u^2=3/4\Rightarrow(u-1/2)^2=0\Rightarrow u=1/2$. So $\cos2\theta=\pm1/\sqrt2$, $2\theta=\pi/4,3\pi/4$ in $(0,\pi)$. $\theta=\pi/8,3\pi/8$.
Sum $=\pi/2$.
Correct Answer: 1