Simple Identities

What is a trigonometric identity?

Trigonometric identities 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?

The two trigonometric identities you must know are
- $\tan θ = \frac{\sin θ}{\cos θ}$
  - This is the identity for tan θ
- $\sin^{2} θ + \cos^{2} θ = 1$
  - This is the Pythagorean identity
  - Note that the notation $\sin^{2} θ$ is the same as ${(\sin θ)}^{2}$
Both identities can be found in the formula booklet
Rearranging the second identity often makes it easier to work with
- $\sin^{2} θ = {1 - \cos}^{2} θ$
- $\cos^{2} θ = {1 - \sin}^{2} θ$

Where do the trigonometric identities come from?

You do not need to know the proof for these identities but it is a good idea to know where they come from
From SOHCAHTOA we know that
- $\sin θ = \frac{opposite}{hypotenuse} = \frac{O}{H}$
- $\cos θ = \frac{adjacent}{hypotenuse} = \frac{A}{H}$
- $\tan θ = \frac{opposite}{adjacent} = \frac{O}{A}$
The identity for $\tan θ$ can be seen by diving $\sin θ$ by $\cos θ$
- $\frac{\sin θ}{\cos θ} = \frac{\frac{O}{H}}{\frac{A}{H}} = \frac{O}{A} = \tan θ$
- This can also be seen from the unit circle by considering a right-triangle with a hypotenuse of 1
  - $\tan θ = \frac{O}{A} = \frac{\sin θ}{\cos θ}$
The Pythagorean identity can be seen by considering a right-triangle on the unit circle with a hypotenuse of 1
- Then (opposite)²+ (adjacent)² = 1
- Therefore $\sin^{2} θ + \cos^{2} θ = 1$
Considering the equation of the unit circle also shows the Pythagorean identity
- The equation of the unit circle is $x^{2} + y^{2} = 1$
- The coordinates on the unit circle are $(\cos θ, \sin θ)$
- Therefore the equation of the unit circle could be written $\cos^{2} θ + \sin^{2} θ = 1$
A third very useful identity is $\sin θ = \cos (90 ° - θ)$ or $\sin θ = \cos (\frac{π}{2} - θ)$
- This is not included in the formula booklet but is useful to remember

How are the trigonometric identities used?

Most commonly trigonometric identities are used to change an equation into a form that allows it to be solved
They can also be used to prove further identities such as the double angle formulae

Examiner Tip

If you are asked to show that one thing is identical (≡) to another, look at what parts are missing – for example, if tan x has gone it must have been substituted

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

Simple Identities (DP IB Maths: AI HL)

Revision Note

Simple 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?

Examiner Tip

Worked example

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Author: Amber