5 1 Fundamental Trig Identities Reciprocal Identities sin

5. 1 Fundamental Trig Identities Reciprocal Identities sin ( ) = 1 csc ( ) = 1 sin ( ) Quotient Identities tan ( ) = sin( ) cos( ) Pythagorean Identities sin 2 ( ) + cos 2 ( ) = 1 1 + tan 2 ( ) = sec 2 ( ) 1 + cot 2 ( ) = csc 2 ( ) cos ( ) = 1 tan ( ) = 1 sec ( ) = 1 cot ( ) = 1 cos ( ) tan ( ) cot ( ) = cos( ) sin ( ) Even-Odd (Negative Angle) Identities sin(-x) = -sin (x) cos(-x) = cos (x) tan(-x) = -tan (x) csc(-x) = -csc(x) sec(-x) = sec(x) cot(-x) = -cot(x)

5. 1 Practice Problems 1. 2. 3. 4. 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) = ______ 5. Find sin(θ), given cos(- θ) = √ 5/5 and tan θ < 0 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 (Hint: Use sin 2 θ + cos 2 θ = 1)

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 – 2 sin 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 - sin cos ( - ) = cos + sin ( + ) = sin cos + cos sin ( - ) = sin cos - cos sin Co-Function Identities sin ( ) = cos (90 - ) cos ( ) = sin (90 - ) tan ( ) = cot (90 - ) cot ( ) = tan (90 - ) sec ( ) = csc (90 - ) csc ( ) = sec (90 - ) tan ( + ) = tan + tan 1 - tan ( - ) = tan - tan 1 + tan

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 2 = cos 2 - sin 2 = 2 cos 2 - 1 = 1 – 2 sin 2 Power Reducing Formulas sin 2 = 1 – cos 2 2 cos 2 = 1 + cos 2 2 tan 2 = 1 – cos 2 1 + cos 2 tan 2 = 2 tan 1 – tan 2 Half Angle sin = 2 1 – cos 2 tan = 2 cos = 2 1 – cos sin = 1 + cos 2 sin = +/1 + cos 1 -cos. A 1+cos. A

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

Double Angle Matching Practice #1 2 cos 2 15◦ - 1 = ____ A. 1/2 #2 B. √ 2 / 2 2 tan 15◦ = _____ 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: 1. Product to Sum/Sum to Product Practice 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 1. Half-Angle Practice Find the exact value of cos 15◦ (sin x + cos x)2 = sin 2 x + 1