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Pre Calculus Chapter 7 Outline

Pre Calculus Chapter 7 Outline. A Presentation By Cody Lee & Robyn Bursch. Section 7.1 : Inverse Sine, Cosine, and Tangent Functions. y= sin x means x= sin y where -1 ≤ x ≤ 1, - π /2 ≤ y ≤ π /2 y= cos x means x= cos y where -1 ≤ x ≤ 1, 0 ≤ y ≤ π

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Pre Calculus Chapter 7 Outline

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  1. Pre CalculusChapter 7 Outline A Presentation By Cody Lee & Robyn Bursch

  2. Section 7.1 : Inverse Sine, Cosine, and Tangent Functions • y= sin x means x= sin y where -1 ≤ x ≤ 1, - π/2 ≤ y ≤ π/2 • y= cos x means x= cos y where -1 ≤ x ≤ 1, 0 ≤ y ≤ π • y= tan x means x= tan y where -∞ < x < ∞, 0 < y < π • y= sec x means x= sec y where |x|≥1, 0 ≤ y ≤ π, y≠ π/2 • y= csc x means x= csc y where |x|≥1, - π/2 ≤ y ≤ π/2, y≠0 • y= cot x means x= cot y where -∞ < x < ∞, 0 < y < π See pg 489 for formulas, pg 429-442 for detailed explanation -1 -1 -1 -1 -1 -1

  3. 7.1 Continued

  4. Section 7.2 : Inverse Trigonometric Functions (Continued)

  5. Ex: Find the exact value of: tan [cos (-1/3)] -1 • tan [cos (-1/3)] • Ѳ= cos (-1/3), so cos Ѳ = -1/3 and since cos Ѳ < O, Ѳ lies in quadrant II • Since cos is x/r, we let x=-1 and r=3 • Use pythagoream theorem to find y • (-1)²+ y² = 3² » 9-1=y² » y²=8 » y=2√2 • Since we have y=2√2 and r= 3, tan [cos (-1/3)] = tan Ѳ= (2√2)/-1 = - 2√2 -1 -1

  6. Section 7.3 : Trigonometric Identities

  7. Ex: Techniques to Simplify Trigonometric Expressions • Show that cosѲ / 1+ sin Ѳ = 1-sin Ѳ /cosѲ by multiplying the numerator and denominator by 1-sin • Solution: cosѲ/ 1+ sin Ѳ = cosѲ/ 1+ sin Ѳ x 1-sin Ѳ/ 1-sin Ѳ • = cosѲ (1-sin Ѳ)/1-sin² Ѳ • = cosѲ (1-sin Ѳ)/cos² Ѳ = 1-sin Ѳ/ cos Ѳ

  8. Section 7.4 : Sum and Difference Formulas

  9. Ex: Using Sum Formula to Find Exact Values • Find the exact value of cos(75°) • Solution: since 75° = 45°+30°, we use the formula for cos(α+β) • Cos 75° = (45°+30°) = cos 45°cos 30° - sin 45°sin 30° • = (√2/2)(√3/2)- (√2/2)( ½ ) • = ¼(√6-√2)

  10. Section 7.5 : Double-Angle Formulas

  11. 7.5: Double-Angle using Squares

  12. Section 7.5: Half-Angle Formulas

  13. Ex: Finding Exact Values Using Double-Angle • If sin Ѳ= 3/5 and π/2 < Ѳ < π, find the exact value of cos (2Ѳ) • Solution: because we are given sin Ѳ= 3/5, we can use the formula cos (2Ѳ)= 1 - 2sin²Ѳ. • cos (2Ѳ)= 1- 2(3/5)² » 1- 2(9/25) » 1- 18/25 • cos (2Ѳ)= 7/25

  14. Ex: Finding Exact Values Using Half-Angle Formulas • Use a half-angle formula to find the exact value of: sin(-15°) • Solution: We use the fact that sin(-15°)= -sin(15°) and 15°= 30°/2 • Use the formula sin α/2= ± √(1 - cos α/2) • sin(-15°) = -sin(30°/2) = - √(1 - cos30°/2) » - √(1 –(√3/2)/2) » 2 (- √(1 –(√3/2)/2) » (- √(2 –√3)/4) • = - √(2 –√3)/2

  15. Thanks for Watching! Good Luck on the Final…

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