Linear Combinations of Sinusoids Dr. Shildneck Fall 2014

Linear Combinations of Sinusoidal Functions 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, 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.

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

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.

Example Write 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. 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.

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

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

v θ = u - v u

θ

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

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

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

SUM and DIFFERENCE IDENTITIES

Example 1 Find the exact value of

Example 2 Find the exact value of

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

Example 4 Write as an expression of x.

Example 5 Solve on

ASSIGNMENT Alternate Text P. 395 #63-84(m3), 85, 87, 93-108(m3) Foerster P. 394 #1-9 (odd), 17, 23, 25