Question

If $\sin^{6}\theta = 1 + \cos^{4}3\theta$ then the most general value of $\theta$ is: (where n is an integer)

• A. $(n+\frac{1}{2})\pi$
• B. $(2n+1)\frac{\pi}{6}$
• C. $(n+\frac{1}{2})\frac{\pi}{2}$

Answer 1

Answer: (A)

$(n+\frac{1}{2})\pi$

Answer 2

For equality, $\sin^6\theta=1$ gives $\theta=\pi/2+n\pi$ and $\cos^4(3\theta)=0$ gives $\theta=\pi/6+(m\pi)/3$. These two are consistent if $m=1+3n$. Hence, $\theta = (n+\frac{1}{2})\pi$.