Trigonometric Identities Unit 5.1

Define Identity If left side equals to the right side for all values of the variable for which both sides are defined. 2. Classic example a2 + b2 = c2 x2 – 9 = x + 3 x ≠ 3 x – 3

Not an Identity x2 = 2x true when x = 0,2 not for other values sinx = 1 – cosx True when x = 0 Sin(0) = 1 – cos(0) or 0 = 1 – 1 Not true when x = π/4 Sin(π/4) ≠ 1 – cos(π/4) or sin√2/2 ≠1 - √2/2

Reciprocal and quotient identities Reciprocal Identities Sinθ = 1/cscθ cscθ =1/sinθ cosθ = 1/secθ secθ =1/cosθ Quotient Identities Tan = sin/cos Cotangent = cos/sin

Diagram

Unit 5.1 Page 312 Guided Practice 1a If sec x = 5/3 find cos x cos = 1/sec cos = 1/(5/3) cos = 3/5 Guided Practice 1b If csc β= 25/7 and sec β= 25/24, find tan β Sin = 1/csc Sin = 1/(25/7) = 7/25 Cos = 1/sec Cos = 1/(25/24) = 24/25 5. Tan = sin/cos = (7/25)/(24/25) tan = 7/24

Unit 5.1 Page 317 Problems 1 - 8 1. if cot θ = 5/7, find tan θ 2. tan = 1/cot 3. tan = 1/(5/7) 4. tan = 7/5

Pythagorean Identities sin2 θ + cos2 θ = 1 0o 02 + 12 = 1 30o .52 + (√3/2)2 = 1 45o (√2/2)2 +(√2/2)2 = 1 60o (√3/2)2 + .52 = 1 90o 12 + 02 = 1

Other Pythagorean Identities tan2 θ + 1 = sec2 cot2 θ + 1 = csc2 θ

Guided practice 2a Csc θ and tan θ, cot θ = -3, cos θ < 0 1. cot2 θ + 1 = csc2 2. (-3) 2 + 1 = csc2 3. 10 = csc2 4. √10 = csc

Guided Practice 2a cont. Csc = 1/sin or √10 = 1/sin √10/10 = sin cot= cos/sin -3 = cos/(√10/10) Cos = (-3√10)/10 Tan = sin/cos Tan = (√10/10)/ (-3√10)/10 Tan = -1/3

Guided Practice 2b Find Cot x and sec x; sin x = 1/6, cos x > 0 Step 1 find sec sin2 + cos2 = 1 (1/6)2 + cos2 = 1 1/36 + cos2 = 1 cos2 = 1 – 1/36 Cos = √35/36 or 1/6√35 Sec = 1/cos or 1/ (1/6√35) or 6 √35/35

Guided Practice 2b Cont. Step 2: Find cot cot = 1/tan Cot = 1/(sin/cos) Cot = 1/(1/6)/(1/6√35) Cot = √35

Unit 5.1 Page 317 Problems 9 - 14