Trigonometric ratios
Trigonometry is used to find 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 (\(x\)).
Three trigonometric ratios
Trigonometry involves three ratios - sine, cosine and tangent which are abbreviated to \(\sin\), \(\cos\) and \(\tan\).
The three ratios are found 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.
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°.
| \(\sin{x} = \frac{o}{h}\) | \(\cos{x} = \frac{a}{h}\) | \(\tan{x} = \frac{o}{a}\) |
| \(\sin{30} = \frac{1}{2}\) | \(\cos{30} = \frac{\sqrt{3}}{2}\) | \(\tan{30} = \frac{1}{\sqrt{3}}~\text{or}~\frac{\sqrt{3}}{3}\) |
| \(\sin{60} = \frac{\sqrt{3}}{2}\) | \(\cos{60} = \frac{1}{2}\) | \(\tan{60} = \sqrt{3}\) |
| \(\sin{x} = \frac{o}{h}\) | \(\sin{30} = \frac{1}{2}\) |
|---|---|
| \(\cos{x} = \frac{a}{h}\) | \(\cos{30} = \frac{\sqrt{3}}{2}\) |
| \(\tan{x} = \frac{o}{a}\) | \(\tan{30} = \frac{1}{\sqrt{3}}~\text{or}~\frac{\sqrt{3}}{3}\) |
| \(\sin{x} = \frac{o}{h}\) | \(\sin{60} = \frac{\sqrt{3}}{2}\) |
|---|---|
| \(\cos{x} = \frac{a}{h}\) | \(\cos{60} = \frac{1}{2}\) |
| \(\tan{x} = \frac{o}{a}\) | \(\tan{60} = \sqrt{3}\) |
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.
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°.
| \(\sin{x} = \frac{o}{h}\) | \(\cos{x} = \frac{a}{h}\) | \(\tan{x} = \frac{o}{a}\) |
| \(\sin{45} = \frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) | \(\cos{45} = \frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) | \(\tan{45} = 1\) |
| \(\sin{x} = \frac{o}{h}\) | \(\sin{45} = \frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) |
|---|---|
| \(\cos{x} = \frac{a}{h}\) | \(\cos{45} = \frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) |
| \(\tan{x} = \frac{o}{a}\) | \(\tan{45} = 1\) |
The accurate trigonometric ratios for 0°, 30°, 45°, 60° and 90° are:
| \(0^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) | \(90^\circ\) | |
| \(\sin{x}\) | \(0\) | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(\cos{x}\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(0\) |
| \(\tan{x}\) | \(0\) | \(\frac{1}{\sqrt{3}}~\text{or}~\frac{\sqrt{3}}{3}\) | \(1\) | \(\sqrt{3}\) | \(\text{Undefined}\) |
| \(\sin{x}\) | |
|---|---|
| \(0^\circ\) | \(0\) |
| \(30^\circ\) | \(\frac{1}{2}\) |
| \(45^\circ\) | \(\frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) |
| \(60^\circ\) | \(\frac{\sqrt{3}}{2}\) |
| \(90^\circ\) | \(1\) |
| \(\cos{x}\) | |
|---|---|
| \(0^\circ\) | \(1\) |
| \(30^\circ\) | \(\frac{\sqrt{3}}{2}\) |
| \(45^\circ\) | \(\frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) |
| \(60^\circ\) | \(\frac{1}{2}\) |
| \(90^\circ\) | \(0\) |
| \(\tan{x}\) | |
|---|---|
| \(0^\circ\) | \(0\) |
| \(30^\circ\) | \(\frac{1}{\sqrt{3}}~\text{or}~\frac{\sqrt{3}}{3}\) |
| \(45^\circ\) | \(1\) |
| \(60^\circ\) | \(\sqrt{3}\) |
| \(90^\circ\) | \(\text{Undefined}\) |
\(\tan{90}\) is undefined because \(\tan{90} = \frac{1}{0}\) and division by zero is undefined (a calculator will give an error message).