1. Linear Combinations of Sinusoids
Dr. Shildneck
Fall 2014

2. Linear Combinations of Sinusoidal Functions
Part I
Linear Combinations of Sinusoidal Functions

3. 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.

7. 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.

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

9. 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. A
b = 2 D
a = 1
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.

10. Example
Write in terms of a single cosine function.

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

12. Sum and Differences Of Periodic Functions
Part II
Sum and Differences Of Periodic Functions

13. Derive the Cosine of a Difference
Using the Unit Circle to
Derive the Cosine of a Difference

14. v
θ = u - v u

15. θ

16. θ
θ = u - v
Since , we can write an equivalence relation for the lengths of the segments.

17. Derive the Cosine of a Sum
Use the previous identity and even/odd identities to
Derive the Cosine of a Sum

18. 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
Derive the Sine of a Sum and The Sine of a Difference

19. SUM and DIFFERENCE IDENTITIES

20. Example 1
Find the exact value of

21. Example 2
Find the exact value of

22. Example 3 Find the exact value of if ,
in Quadrant 1 and in Quadrant 2.

23. Example 4
Write as an expression of x.

24. Example 5
Solve on

25. ASSIGNMENT
Alternate Text P. 395 #63-84(m3), 85, 87, (m3) Foerster P. 394 #1-9 (odd), 17, 23, 25