Linear Combinations of Sinusoids · 2014. 11. 3. · The Linear Combination of Sine and Cosine...

Linear Combinations of Sinusoids

Dr. Shildneck

Fall 2014

LINEAR COMBINATIONS OF SINUSOIDAL FUNCTIONS

Part I

The Sum of Two Sinusoids

If you add two sinusoidal functions (wave functions) that are not in phase, the result will be another sinusoid with the same period and a phase shift.

The amplitude of the result will be less than the sum of the amplitudes of the composed functions.

The linear combination of function,

can be written as a single cosine function with a phase displacement (shift) in the form

Where A = amplitude of the new wave, and

D = the phase displacement.

cos siny a x b x

cos( )y A x D

D

A

Graphically, D is the shift of the sinusoidal curve , while A is the amplitude of the new curve.

cosy x

a = 1

b = 2

D

A

Q1. How can we use this information to find D?

Q2. How can we use it to find A? Use the Pythagorean Theorem.

Use arctan(b/a).

On the unit circle, D is an angle, in standard position whose horizontal component is a and vertical component is b (which are the coefficients in the original combination). A (the amplitude) is the distance of the horizontal and vertical components of the combination.

cos 2sin

Example

Write in terms of a single cosine function.

3cos 4siny

PROPERTY The Linear Combination of Sine and Cosine functions with equal periods, can be written as a single cosine function with phase displacement.

cos sin cos( )a x b x A x D

Where and . 2 2A a b arctanb

Da

Note: The signs of a and b specify the appropriate quadrant for D. A should be written in exact terms when possible. D can be rounded to 3 decimals.

SUM AND DIFFERENCES OF PERIODIC FUNCTIONS

Part II

DERIVE THE COSINE OF A DIFFERENCE

Using the Unit Circle to

θ = u - v

(cos ,sin )B v v

(cos ,sin )A u u

u v

'(1,0)B

'(cos ,sin )A

'(1,0)B

'(cos ,sin )A

θ

θ = u - v

(cos ,sin )B v v

(cos ,sin )A u u

Since , we can write an equivalence relation for the lengths of the segments. ' 'AB A B

DERIVE THE COSINE OF A SUM

Use the previous identity and even/odd identities to

DERIVE THE SINE OF A SUM AND THE SINE OF A DIFFERENCE

You can use the previous identities, co-function identities, and even/odd identities to

sin( ) sin cos sin cos

cos( ) cos cos sin sin

sin( ) sin cos sin cos tan tantan( )

cos( ) cos cos sin sin 1 tan tan

x y x y y x

x y x y x y

x y x y y x x yx y

x y x y x y x y

SUM and DIFFERENCE IDENTITIES

Example 1

Find the exact value of cos75

Example 2

Find the exact value of 7

sin12

Example 3

Find the exact value of if ,

sin( )u v4

sin5

u

in Quadrant 1 and in Quadrant 2. 5

tan12

v

Example 4

Write as an expression of x. cos(arctan1 arccos )x

Example 5

Solve on sin sin 14 4

x x

[0,2 )

ASSIGNMENT

Alternate Text

P. 395 #63-84(m3), 85, 87, 93-108(m3)

Foerster

P. 394 #1-9 (odd), 17, 23, 25