Trig Identities

Trig identities are nothing more than algebraic expressions of the geometry of the unit circle.

For example,

shows us quite simply that:

p = cos(q) = cos(–q)

q = sin(q) = –sin(–q)

These are often called the opposite angle formulas. They tell us how the sine and the cosine will behave when we replace any input with its opposite. In particular, these identities tell us that the cosine is an even function and the sine is an odd function. (See the section on power functions.)

If we apply the Pythagorean Theorem at any point on the unit circle

we see that:

p ² + q ² = [cos(q)] ² + [sin(q)] ² = 1

This is usually called the Pythagorean identity. It tells us that when this combination of sine and cosine appears in our work we can always replace it with the identical, and much simpler, 1 .

Other identities are discovered in a similar way, and similarly help us to simplify our algebraic manipulations of the trigonometric functions. Here are a few more:

• The double angle formulas:

  sin(2q) = 2 sin(q) cos(q)
  
  cos(2q) = [cos(q)] ² – [sin(q)] ²

  If we solve the Pythagorean identity for  [cos(q)] ²  or  [sin(q)] ²  and plug into the cosine double angle formula above, we arrive, respectively, at the following two additional identities:

  cos(2q) = 1 – 2 [sin(q)] ²
  
  cos(2q) = 2 [cos(q)] ² – 1
  
  

• The half angle formulas:

  sin(q/2) =
  
  cos(q/2) =
  
  

• The sum and difference formulas:

  sin(q ± j) = sin(q) cos(j) ± cos(q) sin(j)
  
  cos(q ± j) = cos(q) cos(j) sin(q) sin(j)

What other trig identities can you discover?