Find the integer $n,$ $-90 \le n \le 90,$ such that $\sin n^\circ = \cos 456^\circ.$
I know that the cosine function has a period of 360 degrees, so I can subtract 360 from 456 until I get an angle in the interval $[0, 360).$ So, $\cos 456^\circ = \cos (456 - 360)^\circ = \cos 96^\circ.$ Now I want to find an angle whose sine is equal to $\cos 96^\circ.$ I recall that $\sin \theta = \cos (90 - \theta)$ for any angle $\theta,$ so I can use this identity to find the answer. Let $\theta = n^\circ,$ then $\sin n^\circ = \cos 96^\circ$ implies that $n^\circ = 90 - 96^\circ = -6^\circ.$ This is an integer in the given interval, so I have found the answer. 