6 2 Sum Difference and Double Angle Identities

6. 2 Sum, Difference, and Double Angle Identities The expressions sin (A + B) and cos (A + B) occur frequently enough in math that it is necessary to find expressions equivalent to them that involve sines and cosines of single angles. So…. Does sin (A + B) = Sin A + Sin B Math 30 -1 Let A = 30 and B = 60 1

Sum and Difference Identities Formula Sheet sin (A + B) = sin A cos B + cos A sin B sin (A - B) = sin A cos B - cos A sin B cos (A + B) = cos A cos B - sin A sin B cos (A - B) = cos A cos B + sin A sin B Math 30 -1 2

Simplifying Trigonometric Expressions 1. Express cos 1000 cos 800 + sin 800 sin 1000 as a trigonometric function of a single angle. This expression 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. as a single trig function. Express This expression has the same pattern as sin(A - B), with Math 30 -1 3

Determine Exact Values using Sum or Difference Identities 1. Determine the exact value for sin 750. Think of the angle measures that produce exact values: 300, 450, and 600. Use the sum and difference identities - which angles, used in combination of addition or subtraction, would give a result of 750? sin 750 = sin(300 + 450) = sin 300 cos 450 + cos 300 sin 450 Math 30 -1 4

Finding Exact Values 2. Determine the exact value for cos 150 = cos(450 - 300) = cos 450 cos 300 + sin 450 sin 300 3. Find the exact value for Math 30 -1 5

Determine the exact value of Determine a common denominator Combine terms in numerator Rationalize the denominator or……. Math 30 -1 6

Using the Sum and Difference Identities Prove L. S. = R. S. Math 30 -1 7

Using the Sum and Difference Identities A x y r Math 30 -1 B 4 3 5 2 3 8

Double Angle Identities The identities for the sine and cosine of the sum of two numbers can be used, when the two numbers A and B are equal, to develop the identities for sin 2 A and cos 2 A = cos (A + A) sin 2 A = sin (A + A) = sin A cos A + cos A sin A = cos A - sin A = 2 sin A cos A = cos 2 A - sin 2 A Identities for sin 2 A and cos 2 A: sin 2 A = 2 sin A cos 2 A = cos 2 A - sin 2 A cos 2 A = 2 cos 2 A - 1 cos 2 A = 1 - 2 sin 2 A Math 30 -1 9

Double Angle Identities Express each in terms of a single trig function. a) 2 sin 45° cos 45 ° sin 2 x = 2 sin x cos x sin 2(45 ° ) = 2 sin 45 ° cos 45 ° = sin 90 ° Math 30 -1 b) cos 2 5 - sin 2 5 cos 2 x = cos 2 x - sin 2 x cos 2(5) = cos 2 5 - sin 2 5 = cos 10 10