Trigonometric ratios

Trigonometry involves calculating angles and sides in triangles.

Labelling the sides

The three sides of a right-angled triangle have special names.

The hypotenuse ( h) is the longest side. It is opposite the right angle.

The opposite side ( o) is opposite the angle in question ( x).

The adjacent side ( a) is next to the angle in question ( x).

Pythagorus triangle with Hypotenuse (h), Adjacent (a), Opposite (o) and angle (x degrees)

Three trigonometric ratios

Trigonometry involves three ratios - sine, cosine and tangent which are abbreviated to \sin, \cos and \tan.

The three ratios are calculated by calculating the ratio of two sides of a right-angled triangle.

  • \sin{x} = \frac{\text{opposite}}{\text{hypotenuse}}
  • \cos{x} = \frac{\text{adjacent}}{\text{hypotenuse}}
  • \tan{x} = \frac{\text{opposite}}{\text{adjacent}}

A useful way to remember these is:

s^o_h~c^a_h~t^o_a

Exact trigonometric ratios for 0°, 30°, 45°, 60° and 90°

The trigonometric ratios for the angles 30°, 45° and 60° can be found using two special triangles.

An equilateral triangle with side lengths of 2 cm can be used to find exact values for the trigonometric ratios of 30° and 60°.

An equilateral triangle with equal sides of 2 cm shown split into two right-angled triangles with 60 degree angles.

The equilateral triangle can be split into two right-angled triangles.

An equilateral triangle with equal sides of 2 cm shown split into two right-angled triangles with 60 and 30 degree angles.

Using either of these right-angled triangles, Pythagoras can be used to find the third side of the right-angled triangle.

A right-angled triangle with side lengths of 2 cm and 1 cm. The third side is shown as Root(22 – 12) = root3 cm.

A right-angled isosceles triangle with two sides of length 1 cm can be used to find exact values for the trigonometric ratios of 45°.

A right-angled isosceles triangle with two sides of length 1 cm and two angles of 45 degrees. Pythagoras theorem can be used to calculate the missing length.

Calculate the length of the third side of the triangle using Pythagoras' theorem.

A right-angled isosceles  triangle with side lengths of 1 cm. The third side is shown as Root(12 +12) = root2 cm.

The exact trigonometric ratios for 0°, 30°, 45°, 60° and 90° are:

0^\circ 30^\circ 45^\circ 60^\circ 90^\circ
\sin{x} 0 \frac{1}{2} \frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} 1
\cos{x} 1 \frac{\sqrt{3}}{2} \frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2} \frac{1}{2} 0
\tan{x} 0 \frac{1}{\sqrt{3}}~\text{or}~\frac{\sqrt{3}}{3} 1 \sqrt{3} \text{Undefined}

\tan{90} is undefined because \tan{90} = \frac{1}{0} and division by zero is undefined (a calculator will give an error message).