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Day 62 Sum and Difference Wrap up and Double Angles

Day 62 Sum and Difference Wrap up and Double Angles. Objectives. Use sum and difference formulas to evaluate, simplify, verify and solve trigonometric functions and expressions. Use double angle formulas to evaluate, simplify, verify and solve trigonometric functions and expressions.

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Day 62 Sum and Difference Wrap up and Double Angles

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  1. Day 62Sum and Difference Wrap up and Double Angles

  2. Objectives Use sum and difference formulas to evaluate, simplify, verify and solve trigonometric functions and expressions. Use double angle formulas to evaluate, simplify, verify and solve trigonometric functions and expressions.

  3. Plan for the Day Homework review - Sum and Difference Formulas: verifying and solving equations. 5.4 Page 384 # 11, 23, 27, 55, 57, 59, 61, 69, 71 Evaluate trigonometric expressions using sum and difference formulas Introduce the double angle formulas and their applications Homework • 5.5 Page 394 # 9, 23-27 odd

  4. 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!

  5. Key steps to use… These three steps are key in simplifying and verifying that require the sum and difference formulas: 1. Write in expanded form 2. Substitute known values 3. Simplify

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

  7. 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)

  8. 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?

  9. 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

  10. 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) From our triangles we findcos 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!

  11. Let’s finish this one sin u = 12/13 and u is in quadrant II. cos v = 3/5 in quadrant IV. Find: sin (v – u)

  12. Try this one sin u = 3/5, cos v = - 5/13 and both u and v are in quadrant II. Find: sin (u + v) cos (u + v) tan (u + v)

  13. Double Angles There are many other formulas. We will only study double angles Again, we are going to use these new formulas to evaluate and solve trigonometric expressions and equations.

  14. Double Angles sin 2x = 2 sin x cos x cos 2x = cos2 x – sin2 x = 2cos2 x – 1 = 1 – 2sin2 x

  15. Using Double Angle Formulas to Solve Trigonometric Equations Find all the solutions in the interval [0, 2π). Check the solutions graphically 2 cos x + sin 2x = 0

  16. Solution

  17. Try These sin 2x sinx = cos x cos 2x + sin x = 0

  18. Using double angle formulas to evaluate trigonometric expressions Using a diagram and the double angle formulas, find: sin 2θ cos 2θ tan 2θ sec 2θ csc 2θ cot 2θ 6 θ 8

  19. Try these

  20. Using Formulas to Evaluate Find the exact value of cos 2u, sin 2u, tan 2u given that cos u = 5/13 and 3π/2 < u < 2π. To find the cos 2u let’s look at our identities – which version would be the most simple to use? cos 2x = cos2 x – sin2 x = 2cos2 x – 1 = 1 – 2sin2 x

  21. Find the exact value of cos 2u, sin 2u, tan 2u given that cos u = 5/13 and 3π/2 < u < 2π.

  22. Now find sin 2u sin 2u = 2 sin u cos u Substitute what you know: 2 sin u (5/13) How do we find sin u? Draw a triangle!

  23. Finish the problem

  24. tan 2u We have cos 2u = -119/169 and sin 2u = -120/169. How do we find the tangent? Remember our quotient identities… Let’s check our answer with the Pythagorean identities.

  25. sin2 x + cos2 x = 1 (-120/169)2 + (-119/169)2 = 1 ?

  26. Try this one Find the exact value of cos 2u, sin 2u, tan 2u given that cos u = -2/3 and π/2 < u < π.

  27. Extra Credit Over break, you have the opportunity to earn extra points. To prepare for our next chapter, research and present a derivation (proof) of the Law of Sines and/or the Law of Cosines. Make sure you understand what you write!

  28. Homework 34 5.4 Page 384 # 37-49 odd 5.5 Page 394 # 9, 23-27 odd

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