3.4.2 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