Simple Identities (DP IB Applications & Interpretation (AI)): Revision Note

Simple Identities

What is a trigonometric identity?

• Trigonometric identities are statements that are true for all values of or 

• 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

  • 

    • This is the identity for tan θ

  • 

    • This is the Pythagorean identity

    • Note that the notation is the same as

• Both identities can be found in the formula booklet

• Rearranging the second identity often makes it easier to work with

  • 

  • 

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

  • 

  • 

  • 

• The identity for  can be seen by diving  by 

  • 

  • This can also be seen from the unit circle by considering a right-triangle with a hypotenuse of 1

    • 

• 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 

• Considering the equation of the unit circle also shows the Pythagorean identity

  • The equation of the unit circle is 

  • The coordinates on the unit circle are 

  • Therefore the equation of the unit circle could be written 

• A third very useful identity is or 

  • 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