Cofunctions

In a right triangle the two acute angles add to 90°. Call one x; the other is 90° - x. The side opposite x is the side next to 90° - x, so the sine of one angle is the cosine of the other:

$$\sin\left(90^{\circ}-x\right) = \cos\left(x\right)$$

$$\cos\left(90^{\circ}-x\right) = \sin\left(x\right)$$

The name comes from the Latin complementi sinus, the sine of the complement: the angle that completes 90°. The pairs go on: $\tan\left(90^{\circ}-x\right) = \cot\left(x\right)$, $\sec\left(90^{\circ}-x\right) = \csc\left(x\right)$. In radians the co-angle of x is $\frac{\pi}{2}-x$, so $\cos\left(\frac{\pi}{2}-x\right) = \sin\left(x\right)$.

Check it at 30°: $\sin\left(60^{\circ}\right) = \frac{\sqrt{3}}{2}$, and $\cos\left(30^{\circ}\right) = \frac{\sqrt{3}}{2}$ too. The identities hold at every angle where both sides are defined, not only for the acute angles of a triangle; the chapter on sum and difference formulas proves them.