Trigonometry Notes: Trig Identities

Pythagoraean Identity
Adding multiples of
You should be able to compute any trig function of an angle of the form in terms of and without using a calculator or tables.
It helps with these identities to remember what sine and cosine mean. Given an angle there is a point on the unit cirle corresponding to . is the -coordinate of this point. is the -coordinate of this point. Rather than try to memorize fomulas you should think of how changing the anlge changes the coordinates of the associated point on the unit circle.
- Multiples of
  and correspond to the same point on the unit circle, so sine and cosine don't change.
- Multiples of
  We've seen this before. Adding to takes you to the opposite point on the unit circle. The coordinates of this points are the negatives of the coordinates of the original point.
- Multiples of
  This one is a little trickier. One way to figure these out is to look at the graphs of sine and cosine and think about how you would have o phase shift one to get to the other. Here we will give a different approach. Notice in the picture that the two triangles are related.
  
  The height of the triangle on the left is equal to the width of the triangle on the right, so
  
  Also, the width of the triangle on the left is equal to the height of the triangle on the left, so
  
  I suggest trying to understand the picture rather than trying to memorize the rules, but if you insist on memorizing here's a tip to help: When adding , sine and cosine switch, and sometimes the sign switches too.
  
  When going from Sine to cosine you keep the Same sign.
  
  When going from Cosine to sine you Change the sign.
  
  You can figure out other multiples of by combining the rules we have learned so far:
  - Suppose and is in the 4th quadrant. Find:
    
    Use the Pythagorean identity to find sine from cosine.