Trigonometric Identities

What is a trigonometric identity?

Trigonometric identities are statements about trigonometric functions like $\sin θ$ , $\cos θ$ and $\tan θ$
- They are true for all values of the angle $θ$
- They can be used to help simplify trigonometric equations before solving them
Sometimes you may see identities written with the symbol instead of an equals sign
- This means 'identical to' or 'equivalent to'

What trigonometric identities do I need to know?

You must know these two trigonometric identities:
- $\tan θ = \frac{\sin θ}{\cos θ}$
  - This is the identity for $\tan θ$
- $\sin^{2} θ + \cos^{2} θ = 1$
  - This is sometimes called the Pythagorean identity
  - Note that the notation $\sin^{2} θ$ is the same thing as ${(\sin θ)}^{2}$
The second identity is often used in one of its rearranged forms
- $\sin^{2} θ = 1 - \cos^{2} θ$
- $\cos^{2} θ = {1 - \sin}^{2} θ$

Exam Tip

When asked to show that one thing is equal or identical to another, look at what parts are 'missing'
- This can help you spot which identity to use

Worked example

Show that the equation $2 \sin^{2} θ - \cos θ = 0$ can be written in the form $a \cos^{2} θ + b \cos θ + c = 0$ , where $a$ , $b$ and $c$ are integers to be found with $a > 0$ .

Note that in the 'target' form there is no $\sin θ$ or $\sin^{2} θ$

That means we want to use a substitution to get rid of the $\sin^{2} θ$ in the original form

We can do this using the identity $\sin^{2} θ + \cos^{2} θ = 1$ , rearranged as $\sin^{2} θ = 1 - \cos^{2} θ$

$\begin{array}{rcl} 2 \sin^{2} θ - \cos θ & = & 2 (1 - \cos^{2} θ) - \cos θ \\ = & 2 - 2 \cos^{2} θ - \cos θ \\ = & - 2 \cos^{2} θ - \cos θ + 2 \end{array}$

Substitute that back into the original equation

$\begin{array}{rcl} - 2 \cos^{2} θ - \cos θ + 2 & = & 0 \end{array}$

Multiply both sides of the equation by $- 1$ to make the $\cos^{2} θ$ coefficient positive

This gets the equation into the required form with $a = 2$ , $b = 1$ and $c = - 2$

$2 \cos^{2} θ + \cos θ - 2 = 0$

Trigonometric Identities (Edexcel IGCSE Further Maths)

Revision Note