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 calculated 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°.

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

Equilateral triangle: 2cm x 2cm x 2cm Right angle triangle: sq root 3cm (a) x 1cm (b) x 2cm ©

The length of the third side of the triangle can be calculated using Pythagoras' theorem.

a^2 + b^2 = c^2

a^2 = c^2 - b^2

a^2 = 2^2 - 1^2

a^2 = 3

a = \sqrt{3}

Use the trigonometric ratios to calculate accurate values for the angles 30° and 60°.

\sin{x} = \frac{o}{h} \cos{x} = \frac{a}{h} \tan{x} = \frac{o}{a}
\sin{30} = \frac{1}{2} \cos{30} = \frac{\sqrt{3}}{2} \tan{30} = \frac{1}{\sqrt{3}}
\sin{60} = \frac{\sqrt{3}}{2} \cos{60} = \frac{1}{2} \tan{60} = \sqrt{3}

A square with side lengths of 1 cm can be used to calculate accurate values for the trigonometric ratios of 45°.

Split the square into two right-angled triangles.

Square = 1cm x 1cm x 1cm x 1cm.Triangle = 1cm (a) x 1cm (b) x sq root 2 (c ), 2 known angles of 45 degrees

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

a^2 + b^2 = c^2

c^2 = 1^2 + 1^2

c = \sqrt{2}

Use the trigonometric ratios to calculate accurate values for the angle 45°.

\sin{x} = \frac{o}{h} \cos{x} = \frac{a}{h} \tan{x} = \frac{o}{a}
\sin{45} = \frac{1}{\sqrt{2}} \cos{45} = \frac{1}{\sqrt{2}} \tan{45} = 1

The accurate 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}} \frac{\sqrt{3}}{2} 1
\cos{x} 1 \frac{\sqrt{3}}{2} \frac{1}{\sqrt{2}} \frac{1}{2} 0
\tan{x} 0 \frac{1}{\sqrt{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).