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 ?
- The Pythagorean identity can be seen by considering a right-triangle with a hypotenuse of 1
- Then (opposite)2 + (adjacent)2 = 1
- 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
Show that the equation can be written in the form , where , and are integers to be found.