SOH CAH TOA (right-triangle trigonometry)

What the symbols mean

$\theta$

one of the two acute angles (not the right angle)

$o$

the opposite side: across the triangle from θ

$a$

the adjacent side: next to θ, but not the hypotenuse

$h$

the hypotenuse: the longest side, opposite the right angle

When to use it

Use it in any right-angled triangle when you know an angle and a side and want another side (the height of a tree from its shadow angle, the length of a ramp), or two sides and want an angle (the slope of a roof). Label the sides from the angle you are using first, pick the ratio that links what you know to what you want, then rearrange. With two sides and no angle, Pythagoras gives the third side. For triangles without a right angle, use the law of sines or the law of cosines.

Why it works

All right triangles with the same acute angle are similar, so the ratio of any two of their sides depends only on the angle. Sine, cosine and tangent are names for those three ratios.

Common mistakes

• Labelling sides before choosing the angle. Opposite and adjacent swap when you switch to the other acute angle.
• Using the calculator in the wrong mode: sin 30 in radians is −0.988, not 0.5.
• Writing ${\sin }^{-1}$ as $1/\sin$. The inverse sine gives an angle; $1/\sin \theta$ is a ratio.
• Using SOH CAH TOA in a triangle with no right angle.