Trigonometric Identities (AQA GCSE Further Maths): Revision Note

Trigonometric identities

What is a trigonometric identity?

• Trigonometric identities (trig identity) are statements that are true for all values (of $x$ or $\theta$)

  • 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

  • $\mathbf{tan}\mathit{\theta }\mathbf{\equiv }\frac{\mathbf{sin}\mathbf{\theta }}{\mathbf{cos}\mathbf{\theta }}$

    • This is the identity for tan θ

  • ${\mathbf{sin}}^{\mathbf{2}}\mathit{\theta }\mathbf{+}{\mathbf{cos}}^{\mathbf{2}}\mathit{\theta }\mathbf{\equiv }\mathbf{1}$

    • You may see this referred to as the Pythagorean identity

    • Note that the notation $\sin {}^2\theta$ is the same as ${\left(\sin \theta \right)}^2$

    • Similar for ${\cos }^2\theta$

• 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 $\tan \theta$ can be seen by diving $\sin \theta$ by $\cos \theta$ using their definitions from SOHCAHTOA

  • $\frac{\sin \theta }{\cos \theta }=\frac{\frac{O}{H}}{\frac{A}{H}}=\frac{O}{A}=\tan \theta$

• 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, $\begin{array}{rcl}\sin \theta & =& \frac{O}{H}=\frac{O}{1},\therefore O=\sin \theta \end{array}$ and $\cos \theta =\frac{A}{H}=\frac{A}{1},\therefore A=\cos \theta$

    • And so ${\sin }^2\theta +{\cos }^2\theta = 1$

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)

    • ${\sin }^2\theta = {1-\cos }^2\theta$

    • ${\cos }^2\theta = {1-\sin }^2\theta$