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 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. 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

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
sin2 θ + cos2 θ = 1
0o = 1
30o (√3/2)2 = 1
45o (√2/2)2 +(√2/2)2 = 1
60o (√3/2) = 1
90o = 1

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

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