Trigonometric Identities

What is a trigonometric identity?

Trigonometric identities (trig identity) are statements that are true for all values (of $x$ or $θ$ )
- They are used to help simplify trigonometric equations before solving them
- Sometimes you may see identities written with the symbol ≡
  - This means 'identical to'

What trigonometric identities do I need to know?

There are two trig identities you need to know and use
- $\tan θ \equiv \frac{\sin θ}{\cos θ}$
  - This is the identity for tan θ
- $\sin^{2} θ + \cos^{2} θ \equiv 1$
  - You may see this referred to as the Pythagorean identity
  - Note that the notation $\sin^{2} θ$ is the same as ${(\sin θ)}^{2}$
  - Similar for $\cos^{2} θ$
Both identities are given on the formulae sheet

Where do the trigonometric identities come from?

You do not need to know the proof for these identities
- However it is a good idea to know where they come from
The identity for $\tan θ$ can be seen by diving $\sin θ$ by $\cos θ$ using their definitions from SOHCAHTOA
- $\frac{\sin θ}{\cos θ} = \frac{\frac{O}{H}}{\frac{A}{H}} = \frac{O}{A} = \tan θ$
The Pythagorean identity can be seen by considering a right-triangle with a hypotenuse of 1
- Then using Pythagoras' theorem ( $a^{2} = b^{2} + c^{2}$ , where $a$ is the hypotenuse)
  - $1^{2} = O^{2} + A^{2}$
  - From SOHCAHTOA, $\begin{array}{rcl} \sin θ & = & \frac{O}{H} = \frac{O}{1}, ∴ O = \sin θ \end{array}$ and $\cos θ = \frac{A}{H} = \frac{A}{1}, ∴ A = \cos θ$
  - And so $\sin^{2} θ + \cos^{2} θ = 1$

How are the trigonometric identities used?

Most commonly trig identities are used to rewrite an equation
Rearrangements of the Pythagorean identity are very useful for rewriting equations
- This allows us to write equations in terms of sine or cosine only (making them easier to solve)
  - $\sin^{2} θ = {1 - \cos}^{2} θ$
  - $\cos^{2} θ = {1 - \sin}^{2} θ$

Exam Tip

If you are asked to show that one expression is identical (≡) to another, look for anything that has gone missing!
- e.g. if $\tan x$ is in the original expression but not the 'answer' it must have been replaced by $\frac{\sin x}{\cos x}$

Worked example

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

aa-sl-3-6-1-we-solutions-1

Trigonometric Identities (AQA GCSE Further Maths)

Revision Note

Trigonometric Identities

What is a trigonometric identity?

What trigonometric identities do I need to know?

Where do the trigonometric identities come from?

How are the trigonometric identities used?

Exam Tip

Worked example

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Author: Paul