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Trigonometry Identities: Sum, Difference, Multiple-Angle, and Law of Sines

Explore trigonometry identities involving sums, differences, multiple angles, and the Law of Sines with proofs, applications, practice problems, and solutions. Learn to derive and apply these formulas to solve triangles and angles.

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Trigonometry Identities: Sum, Difference, Multiple-Angle, and Law of Sines

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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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