If you have the equation $$\sin x=\sin\frac{x}{2}$$ and you need to solve for $x$, such that $0\leq x\leq 2\pi$, can you do this without the use of any identities?

Someone asked me this question and I said that the way to solve could be by using the sum identity. First I subtracted $\sin(\frac{x}{2})$ from both sides and then applied the identity: $$-\sin(s)+\sin(t)=-2\cos(\frac{s+t}{2})\sin(\frac{s-t}{2})$$

So I simplified my equation to get $$2\cos(\frac{3x}{4})\sin(\frac{x}{4})=0$$

Then solving $\cos(\frac{3x}{4})=0$ and $\sin(\frac{x}{4})=0$, I get that $x=\frac{2\pi}{3}, 2\pi, 0$

However the person is not familiar with any identities and I wanted to know if there is a way to solve this without them.

  • 1
    $\begingroup$ Note that if sin A = sin B, A=B + 2pik or A = 180 - B + 2pik $\endgroup$ – Gabe Nov 25 '19 at 22:12

We can simply use that by definition

$$\sin(x)=\sin\left(\frac{x}{2}\right) \quad \iff\quad x=\frac x2+2k\pi \quad \lor \quad x=\pi-\frac x2+2k\pi $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.