5-2 Composite Argument and Linear Combination Properties

Hey class. Today we learned about the Linear Combination Property. This property is used to find the amplitude and phase displacement when you combine cosine and sine equations(this is called a Linear Combination).

Linear Combination: b cos(x)+c sin(x)

Linear Combination Property:
b cos(x)+c sin(x) = A cos(x-D)
where
A = (b^2+c^2)^.5 (square root) and
D = arctan(c/b)

You determine the quadrant that the angle is in with A. If D is negative then add 180 degrees. This is because arctan= tan^-1(theta)+180n

We also learned about the Composite Argument Property for Cosine(A-B) which is used to solve cosine equations that have addition or subtration in them.

Composite Argument Property for Cosine (A-B):
cos(A-B) = cos(A)cos(B)+sin(A)sin(B)

Practice Problem
Find the Acos(x-D) equation for y=-8cos(theta)+3sin(theta)
A= square root of (-8^2+3^2) or square root (73)
D= tan^-1(3/-8) = -21 degrees
-21 degrees negative so add 180 degrees to D to get
D=159 degrees

Answer: square root of (73)*cos(theta-159)


For more on these concepts go to:
http://www.keymath.com/x7186.xml

Tight Poem by Sylvia Plath:
http://www.angelfire.com/tn/plath/daddy.html

Debbie I think that you are up next!