Trigonometric Identities (SQA National 5 Maths): Revision Note

Using trigonometric identities

What is an identity?

• An identity can be thought of as a 'stronger form of an equation'

  • An equation like $3x+4=13$ is only true for a particular value of $x$

  • But an identity like ${\cos }^{2}x°+{\sin }^{2}x°=1$ is always true, for any values of $x$

    • The expressions on the two sides of an identity are 'the same' (mathematically identical)

What trigonometric identities do I need to know?

• You need to know and be able to use the following two trigonometric identities:

  • ${\cos }^{2}x°+{\sin }^{2}x°=1$

  • $\tan x°=\frac{\sin x°}{\cos x°}$

How do I use trigonometric identities?

• A question may ask you to simplify a trigonometric expression, or to rewrite it in a different form

• You may need to use substitution

  • For example replacing $\tan x°$ with $\frac{\sin x°}{\cos x°}$ or vice versa

  • $\sin x°\cos x°\tan x°=\sin x°\cancel{\cos x°}\left(\frac{\sin x°}{\cancel{\cos x°}}\right)={\sin }^{2}x°$

• You may need to use algebraic 'tricks' like factorising to allow you to use an identity

  • $3{\sin }^{2}x°+3{\cos }^{2}x°=3\left({\sin }^{2}x°+{\cos }^{2}x°\right)=3\left(1\right)=3$

• In general

  • Look out for places where one side of an identity (or something close to it) appears in a question

    • You may need to replace this with the other side of the identity

    • Some algebra may be needed first to get an exact match

  • Keep an eye on the target form that you are trying to get an expression into

    • This may help you decide which identity needs to be used