Linear Combinations of Sinusoids
Dr. Shildneck
Fall 2014 Part I LINEAR COMBINATIONS OF SINUSOIDAL FUNCTIONS 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, y acos x b sin x can be written as a single cosine function with a phase displacement (shift) in the form
y Acos( x D )
Where A = amplitude of the new wave, and D = the phase displacement.
Graphically, D is the shift of the sinusoidal curve yx cos , while A is the amplitude of the new curve.
D
A On the unit circle, D is an angle, in standard cos 2sin position whose horizontal A component is a and b = 2 vertical component is b (which are the D coefficients in the original a = 1 combination). A (the amplitude) is the distance of the horizontal and vertical components of the combination.
Q1. How can we use this information to find D? Use arctan(b/a).
Q2. How can we use it to find A? Use the Pythagorean Theorem. Example
Write y 3cos 4sin in terms of a single cosine function. PROPERTY The Linear Combination of Sine and Cosine functions with equal periods, can be written as a single cosine function with phase displacement. acos x b sin x A cos( x D )
22 b Where A a b and D arctan . a
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. Part II SUM AND DIFFERENCES OF PERIODIC FUNCTIONS Using the Unit Circle to DERIVE THE COSINE OF A DIFFERENCE B(cos v ,sin v ) v u
θ = u - v
A(cos u ,sin u ) A'(cos ,sin )
B'(1,0) A'(cos ,sin )
B(cos v ,sin v )
θ B'(1,0)
θ = u - v
A(cos u ,sin u )
Since AB A '' B , we can write an equivalence relation for the lengths of the segments. Use the previous identity and even/odd identities to DERIVE THE COSINE OF A SUM You can use the previous identities, co-function identities, and even/odd identities to DERIVE THE SINE OF A SUM AND THE SINE OF A DIFFERENCE SUM and DIFFERENCE IDENTITIES sin(x y ) sin x cos y sin y cos x
cos(x y ) cos x cos y sin x sin y
sin(x y ) sin x cos y sin y cos x tan x tan y tan(xy ) cos(x y ) cos x cos y sin x sin y 1 tan x tan y Example 1
Find the exact value of cos75 Example 2 7 Find the exact value of sin 12 Example 3 4 Find the exact value of sin( uv ) if sin u , 5 5 in Quadrant 1 and tan v in Quadrant 2. 12 Example 4
Write cos(arctan1 arccos x ) as an expression of x. Example 5 Solve sinxx sin 1 on [0,2 ) 44 ASSIGNMENT
Alternate Text P. 395 #63-84(m3), 85, 87, 93-108(m3)
Foerster P. 394 #1-9 (odd), 17, 23, 25