# Solving $\sin 7\phi+\cos 3\phi=0$

The question is find the general solution of this equation:$$\sin(7\phi)+\cos(3\phi)=0$$

I tried to use the "Sum-to-Product" formula, but found it only suitable for $\sin(a)\pm \sin(b)$ or $\cos(a)\pm \cos(b)$. So I tried to expand $\sin 7\phi$ and $\cos 3\phi$, but the equation became much more complicated..

I'm self studying BUT There's nothing about how to solve this type of equations on my textbook..

reeeaaaaally confused now..

• Fortunately, we know how to rewrite a cosine as a sine of a different angle.... – Hurkyl Jun 24 '12 at 10:06

Hint: Use sum to product! $$\sin 7\phi+\sin \left(\frac{\pi}{2}-3\phi \right)=0$$
A very basic fact that most students seem to consider like a magician's trick is that $$\sin \alpha = \cos \left( \frac{\pi}{2} - \alpha \right).$$ By this elementary identity, we can easily solve equations like $$\sin \alpha = \cos \beta.$$