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