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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.

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6.2 Sum, Difference, and Double Angle Identities

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  1. 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 Let A = 30 and B = 60 Math 30-1

  2. 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

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

  4. DetermineExact Values using Sum or Difference Identities 1.Determinethe 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

  5. Finding Exact Values 2.Determinethe exact value for cos 150. cos 150 = cos(450 - 300) = cos 450 cos 300 + sin 450 sin300 3. Find the exact value for Math 30-1

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

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

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

  9. 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 2A and cos 2A. cos 2A = cos (A + A) = cos A cos A - sin A sin A = cos2A - sin2A sin 2A = sin (A + A) = sin A cos A + cos A sin A = 2 sin A cos A Identities for sin 2Aand cos2A: sin 2A= 2sin A cosA cos2A= cos2A- sin2A cos2A= 2cos2A- 1 cos2A= 1 - 2sin2A Math 30-1

  10. Double Angle Identities Express each in terms of a single trig function. a)2 sin 45° cos45 ° b) cos2 5 - sin2 5 cos 2x = cos2 x - sin2x cos 2(5) = cos2 5 - sin2 5 = cos10 sin 2x = 2sin xcosx sin 2(45 ° ) = 2sin 45 °cos45 ° = sin 90 ° Math 30-1

  11. Double Angle Identities Verify the identity L.S = R.S. Math 30-1

  12. Double Angle Identities Verify the identity L.S = R.S. Math 30-1

  13. Identities Prove L.S. = R.S. Math 30-1

  14. Assignment Suggested Questions: Page 306 1, 2, 4, 5, 7, 8a,b,e, 9, 12, 14, 16, 17, 18, 20 Math 30-1

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