21. Sum and Difference Identities

Often you will have the cosine or sin of the sum or difference of two angles. The formulas are:

Examples – Type 1 1. Express cos 1000 cos 800 + sin 800 sin 1000 as a trig function of a single angle. This function has the same pattern as cos (A - B), with A = 1000 and B = 800. cos 100 cos 80 + sin 80 sin 100 = cos(1000 - 800) = cos 200 2. Express as a single trig function. This function has the same pattern as sin(A - B), with

Example – Type 2 Since it says exact we want to use values we know from our unit circle.

Example – Type 2

Example – Type 3 Since it says exact we want to use values we know from our unit circle. 105° is not one there. Think of the angle measures that produce exact values: 300, 450, and 600. Which angles, used in combination of addition or subtraction, would give a result of 1050?

Example – Type 3

Example – Type 3 A little harder because of radians but ask, "What angles on the unit circle can I add or subtract to get negative pi over 12?" hint: 12 is the common denominator between 3 and 4.

Example – Type 3

Example – Type 4 A B x y r 4 2 3 3 5

Example – Type 4 Find cos(A-B)

There are also sum and difference formulas for tangent that come from taking the formulas for sine and dividing them by formulas for cosine and simplifying (since tangent is sine over cosine).