The six trigonometric ratios

A class visits the Qutub Minar. Can a pupil find its height without climbing it? Looking up at the top makes a right triangle: the ground, the tower, and the line of sight. Trigonometry is the maths of exactly this.

The word comes from Greek: tri (three) + gon (sides) + metron (measure). It is the study of how the sides and angles of a triangle are linked.

Name the sides from the angle you are standing at. In a right triangle, pick an acute angle A:

• the hypotenuse is always the side opposite the 90° angle (the longest side);
• the opposite side faces angle A;
• the adjacent side touches angle A and isn't the hypotenuse.

Stand at the other acute angle and the opposite and adjacent sides swap. The hypotenuse never changes.

Three sides make three pairs, and each pair can be written either way up. That gives six ratios:

Also tan A = sin A / cos A and cot A = cos A / sin A.

sin A is one thing, not 'sin times A'. 'sin' on its own means nothing.

Why these ratios are useful: draw a bigger or smaller right triangle with the same angle A. The triangles are similar (Chapter 6), so the sides grow in the same proportion and the ratios stay the same. sin A depends only on the angle, not on the size of the triangle.

Where 'sine' comes from: Aryabhata (AD 500) called the half-chord ardha-jya, shortened to jya or jiva. The Arabic translation kept jiva, the Latin translation turned it into sinus, and that became 'sine'. Aryabhata called the cosine kotijya.