5-2 : Composite Argument and Linear Combination Properties

Section 5-2 : Compostie Argument and Linear Combination Properties
(This sounds scary...it is scary...but its not that scary.)

Property : The Linear Combinations of Cosine an dSine with Equal Periods :

B cos (x) + C sin (x) = A cos (x-D)

A = √(B squared + C squared)

D = Arctan (C/B)

*Note : Arctan is used instead of just inverse tan, because if inverse tan does not lie in the right quadrant you must add 180 degrees or pi depending on whether you are in degree mode or radian mode*

Property : Composite Argument Property for cos (A-B) :

cos (A-B) = Cos A cos B + sin A sin B

How to solve for x (example problem)

problem : 4 cos (x) + 4 sin (x) = 2 [0 , 4π]

solving :

A = √(16 + 16) = √(32)

D = Arctan (4/4)

√(32) cos ( x - .785 ) = 2 divide by √(32) on both sides

cos ( x - .785 ) = 2/√(32) multiply by arccos on both sides

( x - .785 ) = arccos (2/√(32) ) perform arccos

x - .785 = +/- 1.209 + 2πn add .785 to both sides

x = +/- 1.209 + 2πn + .785

x = 1.209 + 2πn + .785

x = 1.994 , 8.227

x = -1.209 + 2πn + .785

x = -.424 , 5.859 , 12.142

*Note : -.424 is in red because it is less than 0 and does not answer the problem. We must add 2π to -.424 in order to find a solution.*

Answers :

1.994 , 8.227 , 5.859 , 12.142

Thats all folks! You never have to learn another thing for the rest of your lives! (I lied about that last part.)

Reminder to MADI (who needs to give me her camera) that she is the next blogger- person- thing-a-majig

Link : http://www.keymath.com/x7186.xml

This is a link to an interactive graph of the linear combination of cosine and sine with equal periods property. Check this out. It helps a lot.

Personalization : I want to thank the Canadian people who came out to wave -- with all five fingers -- for their hospitality. -- George W. Bush