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Half angle and double half angle formulas.pptx
- 1. HALF ANGLE AND DOUBLEANGLE IDENTITIES HALF ANGLE AND DOUBLE ANGLE IDENTITIES VALENTINO T. LABINE VALENTINO T. LABINE
- 2. OBJECTIVES • Identify doubleand half angle identities • Solve problems involving double and half angle identities OBJECTIVES • Identify double and half angle identities • Solve problems involving double and half angle identities
- 3. Special cases ofthe sum and difference formulas for sine and cosine yields what is known as the double-angle identities and the half-angle identities. DOUBLE ANGLE IDENTITIES DOUBLE ANGLE IDENTITIES Special cases of the sum and difference formulas for sine and cosine yields what is known as the double-angle identities and the half-angle identities.
- 4. DOUBLE ANGLE IDENTITIES sin(α + β) = (sin α)(cos β) + (cos α)(sin β) sin (Θ + Θ) = (sin Θ)(cos Θ) + (cos Θ)(sin Θ) sin (2Θ) = 2 sin Θ cos Θ DOUBLE ANGLE IDENTITIES sin (α + β) = (sin α)(cos β) + (cos α)(sin β) sin (Θ + Θ) = (sin Θ)(cos Θ) + (cos Θ)(sin Θ) sin (2Θ) = 2 sin Θ cos Θ
- 5. DOUBLE ANGLE IDENTITIES cos(α + β) = (cos α)(cos β) + (sin α)(sin β) cos (Θ + Θ) = (cos Θ)(cos Θ) + (sin Θ)(sin Θ) cos (2Θ) = cos2 Θ – sin2 Θ DOUBLE ANGLE IDENTITIES cos (α + β) = (cos α)(cos β) + (sin α)(sin β) cos (Θ + Θ) = (cos Θ)(cos Θ) + (sin Θ)(sin Θ) cos (2Θ) = cos2 Θ – sin2 Θ
- 6. DOUBLE ANGLE IDENTITIES tan(α + β) = tan (Θ + Θ) = tan (2Θ) = DOUBLE ANGLE IDENTITIES tan (α + β) = tan (Θ + Θ) = tan (2Θ) =
- 7. EXAMPLES Example 1. sinΘ = 3/5 0 < Θ < π/2 cos Θ = 4/5 tan Θ = 3/4 DOUBLE ANGLE IDENTITIES DOUBLE ANGLE IDENTITIES EXAMPLES 3 5 Θ 4 3 5 Θ 4 Example 1. sin Θ = 3/5 0 < Θ < π/2 cos Θ = 4/5 tan Θ = 3/4 O X Y
- 8. EXAMPLES Example 1. sinΘ = 3/5 cos Θ = 4/5 tan Θ = 3/4 sin (2Θ) = 2sinΘcosΘ = 2(3/5)(4/5) = 24/25 cos(2Θ) = cos2 Θ-sin2 Θ = (4/5)2 -(3/5)2 = 16/25 – 9/25 = 7/25 DOUBLE ANGLE IDENTITIES DOUBLE ANGLE IDENTITIES EXAMPLES Example 1. sin Θ = 3/5 cos Θ = 4/5 tan Θ = 3/4 sin (2Θ) = 2sinΘcosΘ = 2(3/5)(4/5) = 24/25 cos(2Θ) = cos2 Θ-sin2 Θ = (4/5)2 -(3/5)2 = 16/25 – 9/25 = 7/25
- 9. EXAMPLES Example 1. sinΘ = 3/5 cos Θ = 4/5 tan Θ = 3/4 tan(2Θ) = = = = = 24/7 DOUBLE ANGLE IDENTITIES DOUBLE ANGLE IDENTITIES EXAMPLES Example 1. sin Θ = 3/5 cos Θ = 4/5 tan Θ = 3/4 tan(2Θ) = = = = = 24/7
- 10. EXAMPLES Example 2. cosΘ = 5/13, 3π/2 < Θ < 2π. Find sin and tan. sin Θ = -12/13 tan Θ = -12/5 DOUBLE ANGLE IDENTITIES DOUBLE ANGLE IDENTITIES EXAMPLES Θ 5 13 -12 Θ 5 13 -12 Example 2. cos Θ = 5/13, 3π/2 < Θ < 2π. Find sin and tan. sin Θ = -12/13 tan Θ = -12/5 O X Y
- 11. EXAMPLES Example 2. cosΘ = 5/13, 3π/2 < Θ < 2π. Find sin and tan. sin (2Θ) = 2sinΘcosΘ = 2(-12/13)(5/13) sin (2Θ) = -120/164 DOUBLE ANGLE IDENTITIES DOUBLE ANGLE IDENTITIES EXAMPLES Example 2. cos Θ = 5/13, 3π/2 < Θ < 2π. Find sin and tan. sin (2Θ) = 2sinΘcosΘ = 2(-12/13)(5/13) sin (2Θ) = -120/164
- 12. EXAMPLES Example 2. cosΘ = 5/13, 3π/2 < Θ < 2π. Find sin and tan. sin(2Θ) = -120/169 cos(2Θ) = cos2 Θ-sin2 Θ = (5/13)2 – (-12/13)2 = 25/169 – 144/169 cos(2Θ) = -119/169 DOUBLE ANGLE IDENTITIES DOUBLE ANGLE IDENTITIES EXAMPLES Example 2. cos Θ = 5/13, 3π/2 < Θ < 2π. Find sin and tan. sin(2Θ) = -120/169 cos(2Θ) = cos2 Θ-sin2 Θ = (5/13)2 – (-12/13)2 = 25/169 – 144/169 cos(2Θ) = -119/169
- 13. EXAMPLES Example 2. cosΘ = 5/13, 3π/2 < Θ < 2π. Find sin (2Θ) and tan. (2Θ) sin(2Θ) = -120/169 cos(2Θ) = -119/169 tan(2Θ) = = = = 120/119 DOUBLE ANGLE IDENTITIES DOUBLE ANGLE IDENTITIES EXAMPLES 2tanΘ 1 – tan2 Θ sin(2Θ) cos(2Θ) Example 2. cos Θ = 5/13, 3π/2 < Θ < 2π. Find sin (2Θ) and tan. (2Θ) sin(2Θ) = -120/169 cos(2Θ) = -119/169 tan(2Θ) = = = = 120/119 2tanΘ 1 – tan2 Θ sin(2Θ) cos(2Θ)
- 14. EXAMPLES Example 3. Findthe value of 2sin(75°)cos(75°) 2sinΘcosΘ = sin (2Θ) 2sin(75°)cos(75°) = sin (2•75) sin(150) = ½ DOUBLE ANGLE IDENTITIES DOUBLE ANGLE IDENTITIES EXAMPLES 150° 30° 30° 60° 1 2 √2 150° 30° 30° 60° 1 2 √2 Example 3. Find the value of 2sin(75°)cos(75°) 2sinΘcosΘ = sin (2Θ) 2sin(75°)cos(75°) = sin (2•75) sin(150) = ½ O X Y
- 15. sin2 (Θ) = 1– cos(2Θ) 2 sin2 (Θ/2) = 1 – cos (Θ) 2 √sin2 (Θ/2) = √ 1 – cos (Θ) 2 sin (Θ/2) = ± √ 1 – cos (Θ) 2 HALF ANGLE IDENTITIES HALF ANGLE IDENTITIES sin2 (Θ) = 1 – cos(2Θ) 2 sin2 (Θ/2) = 1 – cos (Θ) 2 √sin2 (Θ/2) = √ 1 – cos (Θ) 2 sin (Θ/2) = ± √ 1 – cos (Θ) 2
- 16. cos2 (Θ) = 1+ cos(2Θ) 2 cos2 (Θ/2) = 1 + cos (Θ) 2 √cos2 (Θ/2) = √ 1 + cos (Θ) 2 cos (Θ/2) = ± √ 1 + cos (Θ) 2 HALF ANGLE IDENTITIES HALF ANGLE IDENTITIES cos2 (Θ) = 1 + cos(2Θ) 2 cos2 (Θ/2) = 1 + cos (Θ) 2 √cos2 (Θ/2) = √ 1 + cos (Θ) 2 cos (Θ/2) = ± √ 1 + cos (Θ) 2
- 17. tan(Θ/2) = == = = HALF ANGLE IDENTITIES HALF ANGLE IDENTITIES tan(Θ/2) = = = = =
- 18. tan(Θ/2) = = ADDITIONAL HALFANGLE IDENTITIES ADDITIONAL HALF ANGLE IDENTITIES tan(Θ/2) = =
- 19. EXAMPLES Example 1. Usethe half angle formula to evaluate cos(15°) 15 = Θ/2 30 = Θ cos (Θ/2) = ± √ 1 + cos (Θ) 2 HALF ANGLE IDENTITIES EXAMPLES HALF ANGLE IDENTITIES Example 1. Use the half angle formula to evaluate cos(15°) 15 = Θ/2 30 = Θ cos (Θ/2) = ± √ 1 + cos (Θ) 2
- 20. EXAMPLES Example 1. Usethe half angle formula to evaluate cos(15°) cos (15°/2) = ± √ 1 + cos (30) 2 = ±= ± = ± = ± = HALF ANGLE IDENTITIES EXAMPLES HALF ANGLE IDENTITIES Example 1. Use the half angle formula to evaluate cos(15°) cos (15°/2) = ± √ 1 + cos (30) 2 = ±= ± = ± = ± =
- 21. EXAMPLES Example 2. Findsin(15°) using a half-angle formula. 15 = Θ/2 30 = Θ sin (Θ/2) = ± √ 1 – cos (Θ) 2 HALF ANGLE IDENTITIES EXAMPLES HALF ANGLE IDENTITIES Example 2. Find sin(15°) using a half-angle formula. 15 = Θ/2 30 = Θ sin (Θ/2) = ± √ 1 – cos (Θ) 2
- 22. EXAMPLES Example 2. Findsin(15°) using a half-angle formula. sin (30°/2) = ± √ 1 – cos (30°) 2 = =± =± =√ HALF ANGLE IDENTITIES EXAMPLES HALF ANGLE IDENTITIES Example 2. Find sin(15°) using a half-angle formula. sin (30°/2) = ± √ 1 – cos (30°) 2 = =± =± =√
- 23. EXAMPLES Example 3. Giventhat tan Θ=(8/15) and Θ lies in quadrant III, find the exact value of the following: sin, cos, tan. HALF ANGLE IDENTITIES EXAMPLES HALF ANGLE IDENTITIES Θ 8 15 17 Θ 15 17 8 Example 3. Given that tan Θ=(8/15) and Θ lies in quadrant III, find the exact value of the following: sin, cos, tan.
- 24. EXAMPLES Example 3. Giventhat tan Θ=(8/15) and Θ lies in quadrant III, find the exact value of the following: sin, cos, tan. sin Θ/2 = ± = ± = ± = ± = ± = = ± • = HALF ANGLE IDENTITIES EXAMPLES HALF ANGLE IDENTITIES Example 3. Given that tan Θ=(8/15) and Θ lies in quadrant III, find the exact value of the following: sin, cos, tan. sin Θ/2 = ± = ± = ± = ± = ± = = ± • =
- 25. EXAMPLES Example 3. Giventhat tan Θ=(8/15) and Θ lies in quadrant III, find the exact value of the following: sin, cos, tan. cos Θ/2 = ± = ± = ± = ± = - HALF ANGLE IDENTITIES EXAMPLES HALF ANGLE IDENTITIES Example 3. Given that tan Θ=(8/15) and Θ lies in quadrant III, find the exact value of the following: sin, cos, tan. cos Θ/2 = ± = ± = ± = ± = -
- 26. EXAMPLES Example 3. Giventhat tan Θ=(8/15) and Θ lies in quadrant III, find the exact value of the following: sin, cos, tan. tan Θ/2 = ± = ± = ± = ± = ±√16 = -4 HALF ANGLE IDENTITIES EXAMPLES HALF ANGLE IDENTITIES Example 3. Given that tan Θ=(8/15) and Θ lies in quadrant III, find the exact value of the following: sin, cos, tan. tan Θ/2 = ± = ± = ± = ± = ±√16 = -4
- 27. TRY THIS! 1) Giventhat sinΘ = -4/5 and Θ lies in quadrant IV, find the exact value of cos(Θ/2). 2) Find the exact value of (75°). Answer: 1) -2/√5 2) 2 + √3 HALF ANGLE IDENTITIES HALF ANGLE IDENTITIES TRY THIS! 1) Given that sinΘ = -4/5 and Θ lies in quadrant IV, find the exact value of cos(Θ/2). 2) Find the exact value of (75°).
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Editor's Notes
- #27 -2/square5