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Trigonometric Identities
Unit 5.1
Define Identity
1. 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
1. cos = 1/sec
2. cos = 1/(5/3)
3. cos = 3/5
• Guided Practice 1b• If csc β= 25/7 and
sec β= 25/24, find tan β
1. Sin = 1/csc
2. Sin = 1/(25/7) = 7/25
3. Cos = 1/sec4. Cos = 1/(25/24) = 24/255. 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
1. 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 = sincot= cos/sin-3 = cos/(√10/10)Cos = (-3√10)/10Tan = sin/cosTan = (√10/10)/ (-3√10)/10 Tan = -1/3
Guided Practice 2b
Find Cot x and sec x; sin x = 1/6, cos x > 0Step 1 find sec1. sin2 + cos2 = 12. (1/6)2 + cos2 = 13. 1/36 + cos2 = 14. cos2 = 1 – 1/36 5. Cos = √35/36 or 1/6√356. 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