Chapter 5 Analytic Trigonometry

Chapter 5 – Analytic TrigonometryTopic 1 – Verifying Trig Identities – p. 650- Homework: p. 658 & 659 #1-60 evenTopic 2 – Double Angle Formulas – p. 672- Homework: p. 680 #15-22 evenTopic 3 – Trigonometric Equations (4 Lessons) – p. 693- Homework: p. 703 #7-44 even, p. 704 #45-74 even

Name:Date:Period:Chapter 5: Analytic TrigonometryTopic 1: Verifying Trig IdentitiesFundamental IdentitiesThese should already be in your memory; otherwise this topic will be a struggle.Strive to commit the following relationships to memory.Reciprocal IdentitiesRatio IdentitiesPythagorean IdentitiesUsing Fundamental IdentitiesBy using a mix of the above identities, we will attempt to verify other given identities. To verify an identity, weshow that one side of the equation is equal to the other.General Tips: When possible, ‘anchor’ to cosine and sine.If equations, work with each side independently, starting with the more difficult-looking side.There may be more than one right way, so your work doesn’t have to match mine or a classmates in manycases.If you find yourself down an unhelpful path, restart.Express each of the following as a single term containing one function.1)2)3)4)5)6)

7)8)9)10)11)12)13)14)15)16)17)

Prove these identities to be true1)2)3)4)Factoring can be helpful on the more complicated looking side:5)6)

Name:Date:Chapter 5: Analytic TrigonometryTopic 2: Double Angle FormulasDouble Angle Formulas:1. Ifandterminates in QII, find the exact value of:a.b.c.2. Ifandterminate in QII, find the exact value of:a.b.c.3. Find the exact value of4. Find the exact value ofPeriod:

Verify the Identities:1.2.3.4.6 Page

5.6.7.8.7 Page

Name:Date:Period:Chapter 5: Analytic TrigonometryTopic 3: Trigonometric Equations (DAY 1)Basic Solving Instructions:Solve for θ: cos θ Steps:1) RA 45 2) Negative3) QIIθ 180 - RAθ 180 – 45θ 135 QIIIθ 180 RAθ 180 45θ 225 1) Determine the R.A. where the cos of θ (IGNORE THE SIGN)2) write the sign (look at the isolated equation)3) write the 2 quadrants where cos is negative4) write the rules for each Quadrantθ {135 , 225 }θ 180 - RAθ RAθ 180 RAθ 360 - RA5) Substitute and solve for θ6) Write solution set and check in your calculatorSolve each equation for all values of θ in the interval1) tan θ 12) sin θ 3) cos θ 4) sin θ 5) tan θ -6) cos θ 8 Page

Solve each equation for all values of θ in the interval7) cos θ 8) sin θ 9) tan θ 10) cot θ -11) sec θ -12) csc θ 9 Page

Solve each equation for all values of θ in the intervalFor these questions you will first have to isolate the function. Also, the Reference Angle is not always going tobe a special angle (from the chart) to solve these you will have to use the arc function (ex: tan-1)Ex: 5cos θ 6 35cos θ -3cos θ RA cos-1RA 53.13 RA 53 (always round to the nearest degree, unless otherwise noted)Now you continue with steps 2 – 6.1) 2sin θ 04) 7sin θ 12 82) 4cos θ 05) 6csc θ – 15 23) 3tan θ 5 26) 3(2sec θ 4) 1910 P a g e

Name:Date:Period:Chapter 5: Analytic TrigonometryTopic 3: Trigonometric Equations (DAY 2)Solve each equation below for all values of the angle in the interval 0 θ 360ºSteps:First Factor the equation (you may use the Quadratic formula). Remember when you are factoring, you areNOT solving for θ, you are solving for the function of θ. Such as sin θ or cos θ. You should have two differentlinear equations.Now follow steps 1- 6 for solving linear trig equationsEx: cos2θ 2cosθ 3-3 -3cos2θ 2cosθ – 3 0(cosθ 3) (cosθ – 1) 0cosθ 3 0 cosθ – 1 0cosθ -3cosθ 1X out of range Quadrantial-1 cos θ 1 θ 0, 360θ {0 }2sin2θ – 5sinθ 2 02sin2θ – 4sinθ – 1sinθ 2 02sinθ (sin θ – 2) – 1(sin θ – 2) 0(2sinθ – 1) (sin θ – 2) 02sinθ – 1 0 sin θ – 2 0sin θ sin θ 21) RA 30X out of range2) positive-1 sin θ 13)IIIθ RA θ 180 - RAθ 30 θ 180 -30θ 150θ {30 , 150 }1. sin2θ 7sinθ – 8 02. cos2θ – 7cosθ 12 011 P a g e

3. tan2θ – 6tanθ 164. cos2θ 25cosθ 24 05. sin2θ sin θ cosθ 06. cos θ tanθcosθ12 P a g e

7. 6sin2θ 7sinθ 38. 4cos2θ 21cosθ – 59. 5tan2θ – 33tanθ – 14 010. tan2θ – 6tanθ – 11 013 P a g e

Name:Date:Period:Chapter 5: Analytic TrigonometryTopic 3: Trigonometric Equations (DAY 3)We can use Pythagorean, and Double Angle Identities to solve the equations.Solve each equation for1. cos2 -1such that 0360 .2. cos2 – cos 03. sin2 sin cos4. cos2 -2cos25. cotx tan26. sin2 sin 014 P a g e

7. 2tan tan28. sin2 tan9. cos2 – 3sin 210. cos2 sin 111. sec2 – tan – 1 012. 1 cot2 2csc – 115 P a g e

13. 2cos sec14. cot 3tan15. 2cos2 – sin – 1 016. 4sin2 4cos 516 P a g e

Name:Date:Period:Chapter 5: Analytic TrigonometryTopic 3: Trigonometric Equations ReviewPractice.Solve each equation forsuch that 0360 .1.2.3.4.5.6.17 P a g e

7.8.9.10. 5tan2 3tan 2 11.12.18 P a g e

13. sin2 2sin 3 14. tan2 tan 16. 3sin2 sin 5 4(1 sin )15. 19 P a g e

Chapter 5: Analytic Trigonometry Topic 3: Trigonometric Equations (DAY 2) Solve each equation below for all values of the angle in the interval 0 θ 360º Steps: First Factor the equation (you may use the Q