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5.1 Fundamental Trig Identities

Reciprocal Identities. 5.1 Fundamental Trig Identities. sin (  ) = 1 cos (  ) = 1 tan (  ) = 1 csc ( ) sec () cot () csc (  ) = 1 sec (  ) = 1 cot (  ) = 1 sin ( ) cos () tan (). Quotient Identities.

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5.1 Fundamental Trig Identities

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  1. Reciprocal Identities 5.1 Fundamental Trig Identities sin () = 1 cos () = 1 tan () = 1 csc () sec () cot () csc () = 1 sec () = 1 cot () = 1 sin () cos () tan () Quotient Identities tan () = sin() cot () = cos() cos() sin () Pythagorean Identities Even-Odd (Negative Angle) Identities sin2 () + cos2 () = 1 1 + tan2 () = sec2 () 1 + cot2 () = csc2 () sin(-x) = -sin (x) csc(-x) = -csc(x) cos(-x) = cos (x) sec(-x) = sec(x) tan(-x) = -tan (x) cot(-x) = -cot(x)

  2. 5.1 Practice Problems • If tan (θ) = 2.6 then tan (-θ) = ________ • If cos (θ) = -.65 then cos (-θ) = _______ • If cos (θ) = .8 and sin (θ) = .6 then tan (- θ) = ________ • If sin (θ) = 2/3 then –sin(- θ) = ___________ • Find sin(θ), given cos(- θ) = √5/5 and tan θ < 0 (Hint: Use sin2θ + cos2θ= 1) • Find sin(θ), given sec θ = 11/4 and tan θ < 0 • Find ALL trig functions given cscθ = -5/2 and θ in Quad III

  3. Identity Matching Practice #1 (cos x)/(sin x) = ________ A. sin2 x + cos2 x #2 tan x = _________ B. cot x #3 cos (-x) = ___________ C. sec2 x #4 tan2x + 1 = ______ D. (sin x)/(cos x) #5 1 = ________ E. cos x

  4. More Identity Matching Practice #1 –tan x cos x = ________ A. (sin2 x) / (cos2 x) #2 sec2x - 1 = _________ B. 1 / (sec2 x) #3 (sec x) / (csc x) = ___________ C. sin (-x) #4 1 + sin2x = ______ D. csc2 x – cot 2 x + sin2 x #5 cos2 x = ________ E. tan x

  5. More Practice: Simplify in terms of sine and cosine • Cot θ sin θ • Sec θ cot θ sin θ • (sec θ – 1)(sec θ + 1) • sin2 θ (csc2θ – 1)

  6. 5.2 Verifying Identities Identity – An equation that is satisfied for all meaningful replacements of the variable Verifying Identities – Show/Prove one side of an equation actually equals the other. Example: Verify the identity: cot  + 1 = csc (cos  + sin ) = 1/sin  (cos  + sin ) = (1/sin ) cos  + (1/sin ) sin  = cos  / sin  + sin  / sin  = cot  + 1 We will do other examples in class • Techniques for Verifying Identities • Change to Sine and Cosine • 2. Use Algebraic Skills – factoring • 3. Use Pythagorean Identities • 4. Work each side separately • (Do NOT add to both sides, etc) • 5. For 1 – sin x, try multiplying • numerator & denominator by • 1 + sin x to obtain 1 = sin2 x

  7. Verify the Following Identities • Cot θ / csc θ = cos θ • (1 – sin2β) / cos β = cos β • (Tan α)/(sec α) = sin α • (Tan2α + 1) / sec α = sec α • Cos2θ (tan2θ + 1) = 1 • Sin2 x + tan2 x + cos2 x = sec2 x • Cos x /sec x + sin x/csc x = sec2x - tan2x

  8. Verify more challenging Identities • Tan x + cos x / (1 + sin x) = sec x • Cos4 x – sin4 x = 1 – 2sin2 x • (Sec x – tan x)2 = (1 – sin x)/(1 + sin x) • (cos2 x – sin2 x) / (1 – tan2 x) = cos2 x • (Sec x + tan x)/ (sec x – tan x) = (1 + 2 sin x + sin2 x)/cos2 x • (sec x – csc x)/(sec x + csc x) = (tan x – 1)/(tan x + 1)

  9. 5.3 & 5.4 Sum and Difference Formulas cos ( + ) = cos  cos  - sin  sin  cos ( - ) = cos  cos  + sin  sin  sin ( + ) = sin  cos  + cos  sin  sin ( - ) = sin  cos  - cos  sin  tan ( + ) = tan  + tan  1 - tan  tan  tan ( - ) = tan  - tan  1 + tan  tan  Co-Function Identities sin () = cos (90 - ) cos () = sin (90 - ) tan () = cot (90 - ) cot () = tan (90 - ) sec () = csc (90 - ) csc () = sec (90 - )

  10. 5.3 & 5.4 Practice Problems Find the exact value of each expression: • Cos 15◦ • Cos (5π/12) • Cos 87 cos 93 – sin 87 sin 93 • Sin (75◦) • Sin 40 cos 160 – cos 40 sin 160 • Tan (7π/12) Use Co-function Identities to find θ • Cot θ = tan 25◦ • Sin θ = cos (-30◦) • Csc (3π/4) = sec θ

  11. 5.5 & 5.6 Double & Half Angle and Power Reducing Formulas Double-Angle sin 2 = 2 sin  cos  cos 2 = cos2  - sin2  = 2 cos2  - 1 = 1 – 2 sin2 tan 2 = 2 tan  1 – tan2  Power Reducing Formulas sin2 = 1 – cos 2 2 cos2  = 1 + cos 2 2 tan2  = 1 – cos 2 1 + cos 2 Half Angle sin  = 1 – cos cos  = 1 + cos  2 2 2 2 tan  = 1 – cos  = sin  = +/- 2 sin  1 + cos  1-cosA 1+cosA

  12. Product-to-Sum & Sum to Product Formulas Sum to Product Product to Sum sin  sin  = ½ [cos ( - ) – cos ( + )] cos  cos  = ½ [cos ( - ) + cos ( + )] sin  cos  = ½ [sin ( + ) + sin ( - )] cos  sin  = ½ [cos ( + ) – sin ( - )] sin  + sin  = 2 sin  +  cos  -  2 2 sin  - sin  = 2 sin  -  cos  +  2 2 cos  + cos  = 2 cos  +  cos  -  2 2 cos  - cos  = -2 sin  +  sin  -  2 2

  13. Double Angle Matching Practice #1 2cos2 15◦ - 1 = ________ A. 1/2 #2 2 tan 15◦= _________ B. √2 / 2 1 – tan2 15◦ #3 2 sin 22.5◦ cos 22.5◦ = ___________ C. √3 / 2 #4 cos2 (π/6) - sin2 (π/6) = ___________ D. - √3 #5 2 sin (π/3) cos (π/3) = _____________ E. √3 / 3 #6 2 tan (π/3) = __________________ 1– tan2 (π/3)

  14. More Double-Angle Practice • Find sin θ+ cosθ, given cos 2θ = ¾ with θ in quadrant III • Cos2 15◦ – sin2 15◦ • Verify the Identity: (sin x + cos x)2 = sin 2x + 1 Product to Sum/Sum to Product Practice • Write 4 cos 75◦ sin 25◦ as the sum or difference of 2 functions • Write sin 2θ – sin 4θ as a product of 2 functions Half-Angle Practice • Find the exact value of cos 15◦

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