**SUM AND DIFFERENCE IDENTITIES**

**DOUBLE ANGLE AND HALF ANGLE IDENTITIES**

Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Given the formulas for the Sum Identity, be able to work with them.

2.Given the formulas for the Difference Identity, be able to work with them.

3.Given the formulas for the Double Angle Identity, be able to work with them.

4.Given the formulas for the Half Angle Identity, be able to work with them.

So far we were exposed several useful trigonometric identities. Namely, the

Reciprocal, Quotient, andPythagorean Identities. We will now discuss a few other identities that can also be useful at times.

They DO NOT have to be memorized! However, you must know how to work with them when given to you.Proofs are provided separately for some of the formulas for your information only. If you are interested you might want to look at them. They can be found under the link "Point of Interest 6" in the

Learning Materials#14 in theMyOpenMathcourse.

Sum Identities- see #1 through 3in the "Examples" documentLet

andube any algebraic expression.v

Difference Identities- see #1 through 3in the "Examples" documentLet

andube any algebraic expression.v

Double Angle Identity for Sine- see #4 and 9in the "Examples" documentLet

be any algebraic expression.u

Double Angle Identity for Cosine- see #5 and 6in the "Examples" documentLet

be any algebraic expression.u

Primary Identity:

Secondary Identities(derived from manipulating the primary identity):1. Replace in the primary identity with (from Pythagorean Identity)

then

2. Replace in the primary identity with (from Pythagorean Identity)

then

Double Angle Identity for Tangent:There are other ways to show the

Double Angle Identityfor tangent, however, this is the easiest one and also the one we will use!Let

be any algebraic expression.u

Half-Angle Identity for Sine- see #7 and 9in the "Examples" documentLet

be any algebraic expression.u

Primary Identity:

Secondary Identities(derived from manipulating the primary identity):1. Replace in the primary identity with (from Pythagorean Identity)

then

2. Replace in the primary identity with (from Pythagorean Identity)

thenLet

be any algebraic expression.u

Half-Angle Identity for Cosine- see #8in the "Examples" documentLet

be any algebraic expression.uLet

be any algebraic expression.u

Half-Angle Identity for Tangent:There are other ways to show the

Half Angle Identityfor tangent, however, this is the easiest one and also the one we will use!Let

be any algebraic expression.u

NOTE: The plus or minus sign in the half angle identities depends on the quadrant in which the angle lies.