Relationships of negative angles  You can convince yourself of these through the use of the unit circle and talking about an initial reference angle theta that is negative. For example show that sin(-2 π/3) = -sin( 2 π/3)

 The equation of a circle is x 2 +y 2 =r 2  Because the unit circle has r=1 and each x value is the same as cos( Ө ) and each y value is the same as sin( Ө ) we get the following idea.  This is true not only because of the equation of a circle but because of the Pythagorean Theorem This is called a PYTHAGOREAN IDENTITY

2 other relationships can occur through simple division.  Start with the above statement and divide both sides by sin 2 Ө  Now start with the same thing and divide both sides by cos 2 Ө  These are also called Pythagorean Identities

Cofunctions  SinCos  SecCsc  TanCot  Notice the names of these things can pair themselves up into groups of 2, we refer to them as cofunctions, what follows is the explanation of what cofunctions are and how they are related to one another.

A B C c a b Lets say that angle A=35 degrees. Then the following would be true. Sin(35)=a/c Tan(35)=a/b What would Cos(B), and Cot(B) end up being? Well we should know that B=90-35 or 55 degrees. Therefore; Cos(55)=a/c Cot(55)=a/b Notice that the 2 cofunctions shown (sine/cosine and tangent/cotangent) though they are of different degrees the ratios for sine and cosine are the same and the ratios for tangent and cotangent are the same. Does it make sense that the sin(A)=cos(B)?

 Basically what we are saying is that angle A and angle B are complementary angles so whatever one angle is, the other is 90-(the first angle).  We will refer to A as the angle we start with and B=90-A  This provides the better notations below…

Identities  You know some basic algebraic identities (equivalent statements that allow you to re-write one expression as another without making numerical changes, we only change the appearance and not the numerical value). For example you know that (a+b) 2 =a 2 +2ab+b 2. That is an identity. The difference of perfect squares is another algebraic identity that you are familiar with.  In this course we are going to work with trigonometric identities. Our goal is to take a complicated expression and use substitution and creativity to re-write the expression as a simpler form without changing numerical values. Do not need to write in your notes

Here are some identities that you should be familiar with so far… The cofunction Identities for whatever reason do not show up very often. However we will be adding to this list.

 There are two styles of problems that you most generally come across. One is to simplify an expression. The other takes two expressions one being more complex than the other, sets them equal to one another and asks you to prove that the more complex expression is actually equal to the simpler expression. In all actuality the process for each of the 2 styles is pretty much the same.  The purpose of doing this is so that we acquire an ability to solve trigonometric equations which is often an integral part of any calculus course. Do not need to write in your notes

A helpful hint:  If you get stuck try putting everything into terms of sin and cos.  Look hard for Pythagorean identities or forms of the Pythagorean identities. For example I would like to find in my expressions because it can be replaced with 1.  Other forms that are useful

 We do not recognize any Pythagorean Identities. So transfer everything over into terms of sin and cos.  Hints: Cancel things that cancel…Re-write expressions with common denominators. USE ALGEBRA (nothing cancels)

 Do you recognize any quotient identities?  Does that help any?  Try re-writing as one fraction.

The following questions ask you to prove or verify an identity. Do the same thing as before, but may have to work on both sides of the equation.