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Trigonometric Identities

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 a 2 + b 2 = c 2 x 2 – 9 = x + 3 x ≠ 3 x – 3. Not an Identity.

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Trigonometric Identities

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  1. Trigonometric Identities Unit 5.1

  2. 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

  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

  4. 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

  5. Diagram

  6. 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 312

  7. 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

  8. Pythagorean Identities • sin2θ + cos2θ = 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

  9. Other Pythagorean Identities tan2θ + 1 = sec2 cot2θ + 1 = csc2θ

  10. Guided practice 2a Csc θ and tan θ, cot θ = -3, cos θ < 0 1. cot2θ + 1 = csc2 2. (-3) 2 + 1 = csc2 3. 10 = csc2 4. √10 = csc

  11. 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

  12. 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

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

  14. Unit 5.1 Page 317 Problems 9 - 14

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