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Day 61 Sum and Difference

Day 61 Sum and Difference. Objectives. To use two new identities to evaluate, simplify, verify, and solve trigonometric equations. The new identities are the sum and difference formulas and double angle formulas. Plan for the Day. Review of homework: 5.3 Page 376 # 19, 20, 33-38 all

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Day 61 Sum and Difference

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  1. Day 61Sum and Difference

  2. Objectives To use two new identities to evaluate, simplify, verify, and solve trigonometric equations. The new identities are the sum and difference formulas and double angle formulas

  3. Plan for the Day Review of homework: 5.3 Page 376 # 19, 20, 33-38 all What is next… Still verifying and solving with new identities Some new applications Homework 5.4 Page 384 # 11, 23, 27, 37-49 odd, 55, 57, 59, 61, 69, 71

  4. Sum and Difference Formulas Evaluate this on your calculator: cos (π/3 + π/6) cos π/3 + cos π/6 Are they equal? The sum of the cosine of the angles (a ratio) does not equal the cosine of the sum of the angles! Try the same with sine.

  5. Historical Note (page 381 of your book) Hipparchus (160 BC) is credited with the invention of trigonometry. He also credited with deriving the sum and difference formula for sine and cosine used in trigonometry today.

  6. Sum and Difference Formula Given u and v are angles: • sin (u + v) = sin u cosv + cos u sin v • sin (u – v) = sin u cos v – cos u sin v • cos (u + v) = cos u cos v – sin u sin v • cos (u – v) = cos u cos v + sin u sin v There is a tangent formula but tangent is just the sine divided by the cosine!

  7. Uses… • Evaluate (two situations) • Simplify or Verify Equations • Solving Equations

  8. Evaluating … Finding the Exact Values… Find the exact value of cos 75o cos 75o = cos (45o +30o) cos (45o +30o) = cos (45o)cos (30o) – sin (45o) sin (30o) These are on the unit circle so easy enough to simplify! We will talk about another “evaluation” at the end of this section.

  9. Try these: cos 165o 165o = 135o + 30o sin 7π/12 7π/12 = π/3 + π/4

  10. Uses – Simplifying or Verifying Identities Remember this identity? sin (π/2– x) = cos x Use the sum and difference formulas to verify. 1. Write in expanded form sin (π/2– x) = sin (π/2)cos (x) – cos (π/2) sin (x) 2. Substitute known values • cos (x) – (0) sin (x) 3. Simplify cos (x)

  11. Key steps to use… These three steps are key in verifying and solving equations that require the sum and difference formulas: 1. Write in expanded form 2. Substitute known values 3. Simplify Sometimes these steps must be repeated…

  12. Try These Simplify cos (θ – 3π/2) Verify

  13. Try one more sin (x + y) sin (x – y) = sin2 x – sin2 y

  14. Using Sums and Difference Formulas to Solve Trigonometric Equations Find all the solutions in the interval [0, 2π). sin (x + π/4) + sin (x – π/4) = -1 We are going to use our three steps • Expand • Substitute known values • Simplify Once we simplify, we must find the angles that make the statement true.

  15. Find all the solutions in the interval [0, 2π). sin (x + π/4) + sin (x – π/4) = -1 Expand: sin (x + π/4) = sin x cos π/4 + cos x sinπ/4 sin (x – π/4) = sin x cos π/4 – cos x sinπ/4 sin x cos π/4 + cos x sinπ/4 + sin x cos π/4 – cos x sinπ/4 = -1 sin x cos π/4+ cos x sinπ/4 + sin x cos π/4– cos x sinπ/4 = -1 2sin x cos π/4 = -1

  16. Find all the solutions in the interval [0, 2π). sin (x + π/4) + sin (x – π/4) = -1 Expand: 2sin x cos π/4 = -1

  17. Try This Find all the solutions in the interval [0, 2π). sin (x + π/6) – sin (x – π/6) = ½

  18. Solving Trigonometric Equations • Expand using sum and difference formulas • Simplify where possible • Substitute known values from the unit circle • Simplify • Find solutions

  19. Using Formulas to Evaluate Find the exact value of the given function, given that: • sin u = 12/13 and u is in quadrant II. • Cos v = 3/5 in quadrant IV. Find : • cos (u + v) • sin (v – u)

  20. sin u = 12/13 and u is in quadrant II. Cos v = 3/5 in quadrant IV.Find cos (u + v) Expand: cos u cos v – sin v sin u Substitute what you know: cos u (3/5)– sin v (12/13) How can we find the other values to complete the problem?

  21. sin u = 12/13 and u is in quadrant II. Cos v = 3/5 in quadrant IV.Find cos (u + v) cos u (3/5)– sin v (12/13) Draw your triangle to calculate the other values • remember to put them in the correct quadrants and … • use the proper signs based upon those quadrants

  22. sin u = 12/13 and u is in quadrant II. Cos v = 3/5 in quadrant IV.Find cos (u + v) cos u sin v – (3/5)(12/13) Cos u = -5/13 …. Sin v = -4/5 (-5/13) (3/5)– (-4/5) (12/13) -15/65 + 48/65 = 33/65 Note: you will always have common denominators, if you don’t you have done something wrong!

  23. Try the next one sin u = 12/13 and u is in quadrant II. Cos v = 3/5 in quadrant IV. Find sin (v – u)

  24. Homework 33 5.4 Page 384 # 11, 23, 27, 37-49 odd, 55, 57, 59, 61, 69, 71

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