5.1 Fundamental Trig Identities sin (  ) = 1cos (  ) = 1tan (  ) = 1 csc (  )sec (  )cot (  ) csc (  ) = 1sec (  ) = 1cot (  ) = 1 sin (  )cos (  )tan (  ) tan (  ) = sin(  )cot (  ) = cos(  ) cos(  ) sin (  ) sin 2 (  ) + cos 2 (  ) = tan 2 (  ) = sec 2 (  ) 1 + cot 2 (  ) = csc 2 (  ) Reciprocal Identities Quotient Identities Pythagorean IdentitiesEven-Odd (Negative Angle) Identities sin(-x) = -sin (x)csc(-x) = -csc(x) cos(-x) = cos (x)sec(-x) = sec(x) tan(-x) = -tan (x)cot(-x) = -cot(x)

5.1 Practice Problems 1.If tan (θ) = 2.6 then tan (-θ) = ________ 2.If cos (θ) = -.65 then cos (-θ) = _______ 3.If cos (θ) =.8 and sin (θ) =.6 then tan (- θ) = ________ 4.If sin (θ) = 2/3 then –sin(- θ) = ___________ 5.Find sin(θ), given cos(- θ) = √5/5 and tan θ < 0 (Hint: Use sin 2 θ + cos 2 θ = 1) 6.Find sin(θ), given sec θ = 11/4 and tan θ < 0 7.Find ALL trig functions given csc θ = -5/2 and θ in Quad III

Identity Matching Practice #1 (cos x)/(sin x) = ________A. sin 2 x + cos 2 x #2 tan x = _________B. cot x #3 cos (-x) = ___________C. sec 2 x #4 tan 2 x + 1 = ______D. (sin x)/(cos x) #5 1 = ________ E. cos x

More Identity Matching Practice #1 –tan x cos x = ________A. (sin 2 x) / (cos 2 x) #2 sec 2 x - 1 = _________B. 1 / (sec 2 x) #3 (sec x) / (csc x) = ___________C. sin (-x) #4 1 + sin 2 x = ______D. csc 2 x – cot 2 x + sin 2 x #5 cos 2 x = ________ E. tan x

More Practice: Simplify in terms of sine and cosine 1.Cot θ sin θ 2.Sec θ cot θ sin θ 3.(sec θ – 1)(sec θ + 1) 4.sin 2 θ (csc 2 θ – 1)

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 1.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 = sin 2 x

Verify the Following Identities 1.Cot θ / csc θ = cos θ 2.(1 – sin 2 β) / cos β = cos β 3.(Tan α)/(sec α) = sin α 4.(Tan 2 α + 1) / sec α = sec α 5.Cos 2 θ (tan 2 θ + 1) = 1 6.Sin 2 x + tan 2 x + cos 2 x = sec 2 x 7.Cos x /sec x + sin x/csc x = sec 2 x - tan 2 x

Verify more challenging Identities 1.Tan x + cos x / (1 + sin x) = sec x 2.Cos 4 x – sin 4 x = 1 – 2sin 2 x 3.(Sec x – tan x) 2 = (1 – sin x)/(1 + sin x) 4.(cos 2 x – sin 2 x) / (1 – tan 2 x) = cos 2 x 5.(Sec x + tan x)/ (sec x – tan x) = (1 + 2 sin x + sin 2 x)/cos 2 x 6.(sec x – csc x)/(sec x + csc x) = (tan x – 1)/(tan x + 1)

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 -  )

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

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

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

Double Angle Matching Practice #1 2cos 2 15 ◦ - 1 = ________A. 1/2 #2 2 tan 15 ◦ = _________B. √2 / 2 1 – tan 2 15 ◦ #3 2 sin 22.5 ◦ cos 22.5 ◦ = ___________C. √3 / 2 #4 cos 2 (π/6) - sin 2 (π/6) = ___________D. - √3 #5 2 sin (π/3) cos (π/3) = _____________E. √3 / 3 #6 2 tan (π/3) = __________________ 1– tan 2 (π/3)

More Double-Angle Practice 1.Find sin θ+ cos θ, given cos 2θ = ¾ with θ in quadrant III 2.Cos 2 15 ◦ – sin 2 15 ◦ 3.Verify the Identity: (sin x + cos x) 2 = sin 2x + 1 Product to Sum/Sum to Product Practice 1.Write 4 cos 75 ◦ sin 25 ◦ as the sum or difference of 2 functions 2.Write sin 2θ – sin 4θ as a product of 2 functions Half-Angle Practice 1.Find the exact value of cos 15 ◦