How to find an angle using a cofunction identity

Key concept: When a sine value is set equal to a cosine value, the cleanest move is to rewrite one side using a cofunction identity. That turns the problem into a standard angle match.

Step by step

We start with

$\sin \frac{2\pi}{5}=\cos x$

Use the cofunction identity

$\sin \theta=\cos\left(\frac{\pi}{2}-\theta\right)$

So

$\sin \frac{2\pi}{5}=\cos\left(\frac{\pi}{2}-\frac{2\pi}{5}\right)$

That means

$\cos x=\cos\left(\frac{\pi}{2}-\frac{2\pi}{5}\right)$

Now simplify the angle:

$\frac{\pi}{2}-\frac{2\pi}{5}=\frac{5\pi-4\pi}{10}=\frac{\pi}{10}$

So one valid angle is

$x=\frac{\pi}{10}$

If the problem asks for the measure of an angle, this is the standard answer in the principal range. In degrees, that is $18^\circ$.

Pitfall alert

A common slip is switching the identity the wrong way around. For example, writing $\sin\theta=\cos\theta$ is not a cofunction identity. You need the complementary-angle form $\sin\theta=\cos(\frac\pi2-\theta)$. Also, check whether your class wants radians or degrees before you stop.

Try different conditions

If the equation were $\sin\frac{2\pi}{5}=\cos x$ over all angles, then cosine’s symmetry gives infinitely many solutions:

$x=\pm \frac{\pi}{10}+2k\pi,\quad k\in\mathbb{Z}$

So the single answer $\frac{\pi}{10}$ is the principal value, while the full family comes from cosine periodicity.

Further reading

cofunction identity, complementary angles, trigonometric equations