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Pre Calculus Chapter 7 Outline

This comprehensive outline presents key concepts related to inverse trigonometric functions, identities, and formulas as part of Pre-Calculus Chapter 7. It covers the definitions of inverse sine, cosine, tangent, secant, cosecant, and cotangent functions, including their applicable ranges. The outline further addresses fundamental techniques for simplifying trigonometric expressions using identities and provides examples of sum and difference formulas, double-angle, and half-angle formulas. Essential step-by-step solutions enable students to grasp these concepts effectively while preparing for their finals.

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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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