Choosing a trig ratio

Objective. Select sine, cosine or tangent from labelled sides.

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Learn: the key idea

Choosing a trig ratio means picking sine, cosine or tangent based on which two sides of a right-angled triangle you know or want, relative to a marked angle. Label the sides hypotenuse (opposite the right angle), opposite (across from the angle) and adjacent (next to the angle, not the hypotenuse). Then SOH-CAH-TOA tells you sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. For example, if you know the hypotenuse and want the opposite side, use sine. Always label sides first before choosing a ratio, since swapping opposite and adjacent gives a wrong equation.

Worked example

In a right-angled triangle, angle A = 40°, the hypotenuse is 10 cm, and you need the side opposite A. Which ratio should you use?

Options

sine
cosine
tangent
Pythagoras

1 $Angle A is known, hypotenuse known, opposite side wanted$
2 $Ratio linking opposite and hypotenuse is sine$
3 $sin 40° = opposite/10$
4 $So the ratio used is sine (sin)$

Answer: sine

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Common mistake. Students often label the adjacent side as the hypotenuse because it looks longest in a sketch; the hypotenuse is always opposite the right angle, so check that first, not the drawing's apparent lengths.

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