Right-triangle ratios

For a chosen acute angle \(\theta\), label the opposite side, adjacent shorter side and hypotenuse. Similar right triangles with the same \(\theta\) have the same side ratios, so the ratios depend on the angle rather than the size of the triangle.

\(\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}},\quad\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}},\quad\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}\). Choose the ratio containing the known side and the required side. Set the calculator to degree mode.

Example

A right triangle has hypotenuse 10 cm and an acute angle of \(30^\circ\). The opposite side is \(10\sin30^\circ=5\text{ cm}\). If an acute angle has opposite side 7 cm and adjacent side 10 cm, then \(\theta=\tan^{-1}(\dfrac7{10})\approx35.0^\circ\).

The hypotenuse is always opposite the right angle. “Opposite” and “adjacent” change when you choose the other acute angle. Here \(\tan^{-1}\) means inverse tangent, not the reciprocal of tangent.