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5.3 Sum and Difference Identities

5.3 Sum and Difference Identities. HW: PG. Pg. 468-469 #1-20 e, 24. Cosine of a Difference Identity. Using the cosine-of-a-difference identity. Find the exact value of cos15° without using a calculator. Cosine of a Sum. Cos(u + v) =. Cosine of a Sum or Difference.

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5.3 Sum and Difference Identities

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  1. 5.3 Sum and Difference Identities HW: PG. Pg. 468-469 #1-20 e, 24

  2. Cosine of a Difference Identity

  3. Using the cosine-of-a-difference identity • Find the exact value of cos15° without using a calculator.

  4. Cosine of a Sum • Cos(u + v) =

  5. Cosine of a Sum or Difference • Cos (u v) = cosucosv sinusinv • The sign switches in either case**

  6. Confirming cofunction Identities • Prove the identities: • Cos((π/2) - x) = sinx • Sin((π/2) - x) = cosx

  7. Sine of a Difference or Sum • Sin(u + v) =

  8. …Difference • Sin(u - v)

  9. Sine of a Sum or Difference • Sin(u v) = sinucosv cosusinv • Signs do not switch in either case***

  10. Using the sum/difference formulas • Write each as the sine or cosine of an angle: • Sin22ºcos13º + cos22ºsin13º • Cos(π/3)cos(π/4) + sin(π/3)sin(π/4) • Sinxsin2x - cosxcos2x

  11. Proving Reduction Formulas • Sin(x + π) = -sinx • Cos(x + 3π/2) = sinx

  12. Tangent of a Difference or Sum • Tan(u v) =

  13. Prove the reduction formula: tan(θ - (3π/2)) = -cotθ

  14. 5.4 - Multiple-Angle Identities HW: Pg. 475 #12-40e

  15. Double-Angle Identities • Sin2u = 2sinucosu • Cos2u = cos2u - sin2u • =2cos2u -1 • =1 - 2sin2u • Tan2u = 2tanu/(1 - tan2u)

  16. Prove the identity: • Sin2u = 2sinucosu

  17. Power-Reducing Identities • Sin2u = (1-cos2u)/2 • Cos2u = (1 + cos2u)/2 • Tan2u = (1 - cos2u)/(1 +cos2u)

  18. Prove the Identity: • Cos4 - sin4 = cos2

  19. Reducing a power of 4 • Rewrite cos4x in terms of trig functions with no power greater than 1.

  20. Finding the Sine of Half an Angle (work with partner) • Recall sin2u = (1 - cos2u)/2 • Use the power-reducing formula to show that sin2 (π/8) = (2 - √2)/4. • Solve for sin(π/8) . Do you take the positive or negative square root? Why? • Use the power-reducing formula to show that sin2 (9π/8) = (2 - √2)/4 • Solve for sin (9π/8) . Do you take the positive or negative square root? Why?

  21. Half-Angle Identities Sin u/2 = Cos u/2 = Tanu/2 =

  22. Solve algebraically in the interval [0,2π): • Sin2x = cosx • (use a double-angle identity)

  23. Using half-angle Identities • Sin2x = 2sin2(x/2)

  24. Practice with a Partner • Pg. 475 #1-39 odd

  25. DO NOW: • Find all solutions to the equation in the interval [0,2п) • Sin2x = 2sinx

  26. 5.5 - The Law of Sines HW: Pg. 484 #1-18e

  27. Deriving the Law of Sines

  28. Law of Sines • In any ∆ABC with angles A, B, and C opposite sides a, b, and c, then: • SinA / a = SinB / b = SinC / c

  29. Solving a triangle given two angles and a side • Solve ∆ABC given that A = 36, B = 48, and a = 8. Solving Triangles (AAS, ASA)

  30. Determining the Number of Triangles EXPLORATION 1: Pg. 480 (work with partner) 1. 2. 3. 4. 5.

  31. Solving a triangle given two sides and an angle • Solve ∆ABC given that a = 7, b = 6, and A = 26.3

  32. Handling the ambiguous case • Solve ∆ABC given that a = 6, b = 7, and A = 30

  33. Application • Locating a Fire Forest Ranger Daniel at ranger station A sights a fire in the direction 32 east of north. Ranger Preshan at ranger station B, 10 miles due east of A, sights the same fire on a line 48 west of north. Find the distance from each ranger station to the fire.

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