Sum and Differences Of Periodic Functions

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Presentation transcript:

Sum and Differences Of Periodic Functions Dr. Shildneck Spring, 2015

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

Given two angles, u and v, we want to find a formula for the cosine of the difference between u and v. v θ = u - v u

θ

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

Derive the Cosine of a Sum Now… we can use the previous identity and the even/odd identities to Derive the Cosine of a Sum And…

Derive the Sine of a Sum and The Sine of a Difference Then… we 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 And…

Derive the Tangent of a Sum and The Tangent of a Difference And then… we can use the previous identities, quotient identities, and even/odd identities to Derive the Tangent of a Sum and The Tangent of a Difference But… we aren’t going to… So, here are the rest…

SUM and DIFFERENCE IDENTITIES sin(1st)cos(2nd) [same operation] sin(2nd)cos(1st) cos(1st)cos(2nd) [opposite operation] sin(1st)sin(2nd)

Example 1 Find the exact value of C) Find the exact value of

Example 2 Find the exact value of

Example 3 Simplify the expression:

Example 4 Simplify the expression:

Example 5 Write as an expression of x.

Example 6 Find the exact value of if , in Quadrant 1 and in Quadrant 2.

ASSIGNMENT Assignment 2 WS