Right-triangle sine, cosine and tangent

Similar triangles give fixed ratios

All right triangles with the same acute angle are similar, so their corresponding side ratios match. Relative to an acute angle θ, the opposite side is across from it, the adjacent side is the other shorter side, and the hypotenuse is opposite the right angle.

Choosing the other acute angle swaps opposite and adjacent, but leaves the hypotenuse unchanged. Label the chosen angle first.

Find a side

A right triangle has hypotenuse 12 cm and an acute angle of 35°. The side opposite that angle is 12 sin 35° ≈ 6.88 cm. Its adjacent shorter side is 12 cos 35° ≈ 9.83 cm. Both must be shorter than 12 cm.

If the adjacent side is 8 cm and the angle 40°, the opposite side is 8 tan 40° ≈ 6.71 cm. The known and wanted sides select tangent; the hypotenuse is not supplied or needed.

Find an angle with the inverse operation

If opposite = 5 and adjacent = 12, then tan θ = 5/12, so θ = tan⁻¹ (5/12) ≈ 22.6°. On a calculator, use inverse tangent in degree mode. Here tan⁻¹ names the inverse function, not 1/tan θ.

An angle of elevation is measured upward from a horizontal; an angle of depression downward from a horizontal. A labelled diagram must identify that horizontal before applying a triangle ratio.

Check your understanding 1 The angle chooses the side names

In a right triangle, the side opposite angle A is 7 cm and the hypotenuse is 10 cm. Find A to one decimal place.

Your answer or drawing on paper

Show answer

sin A = 7/10, so A = sin⁻¹ (0.7) ≈ 44.4°. Cosine would use the adjacent side instead. Check degree mode and that the result is acute.

Transfer to a height difference

From a point 20 m horizontally from a vertical pole, the angle of elevation to its top is 32°. The measuring instrument is 1.5 m above the pole’s base level. Find the modelled pole height.