1 / 14

Mastering Sum, Difference, and Double Angle Identities

Understand and apply sine, cosine, and double angle identities in trigonometry to simplify expressions and find exact values.

gmcarthur
Download Presentation

Mastering Sum, Difference, and Double Angle Identities

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related