Exact Values Some special values of Sin Cos

Exact Values Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values Use Pythagoras to find the 3 rd side 60º 2 2 60º 3 30º 2 60º 1 This triangle will provide exact values for sin, cos and tan 30º and 60º

Exact Values 3 30º 2 Using SOH CAH TOA we can work out the Sin, Cos or Tan of these angles and leave the answer as an exact value 60º 1 Now work out the values for Tan 30, Sin 60, Cos 60 and Tan 60.

Exact Values The table so far x 0º 30º Sin xº ½ Cos xº 3 2 Tan xº 45º 60º 3 2 ½ 3 90º

Exact Values Consider the square with sides 1 unit 1 1 Again use Pythagoras to find the 3 rd side 2 45º 1 Now using this triangle calculate exact values for sin, cos and tan of 45 o

Exact Values The table so far x 0º 30º Sin xº ½ Cos xº 3 2 Tan xº 45º 1 2 1 60º 3 2 ½ 3 90º

Exact Values To fill in the rest of the values we use the trig graphs y=tan x y=sin x 1 0 90 o -1 180 270 o o 360 o 0 y=cos x 1 0 -1 90 o 180 o 270 o 360 o

Exact Values The completed table x 0º 30º 0 ½ Cos xº 1 3 2 Tan xº 0 Sin xº 45º 1 2 1 60º 3 2 90º 1 ½ 0 3 Undefined

Exact Values tan 150 o = - tan(180 - 150) o = tan 30 o (Q 2 so tan neg) cos 225 o = - cos(225 - 180) o = - cos 45 o (Q 3 so cos neg) sin 120 o = sin(180 - 120) o = sin 60 o (Q 2 so sin pos) tan 300 o = - tan(360 -300)o = - tan 60 o (Q 4 so tan neg) Sin +ve (180 o – xo) Tan +ve All +ve (xo) Cos +ve (180 o + xo) (360 o – xo)

Exercise 1 Find the exact value of: 05 -Jan-22 Exact Values 1. Sin 2400 6. Sin 3000 2. Cos 1200 7. Cos 3150 3. Tan 1200 8. Sin 2250 4. Sin 1350 9. Tan 2400 5. Tan 12250 10. Cos 1350

Exact Values Examples Convert the following to radians:

Exact Values Exercise 2 Convert to radians: 05 -Jan-22 1. 1800 6. 3000 2. 1500 7. 450 3. 900 8. 2600 4. 1350 9. 3300 5. 2250 10. 3150

Exact Values Examples Convert the following to degrees:

Exercise 3 Convert to degrees: 05 -Jan-22 Exact Values

Exact Values The completed exact values table for angles measured in radians. x 0 0 ½ Cos x 1 3 2 Tan x 0 Sin x 1 2 1 3 2 1 ½ 0 3 Undefined

Exact Values When working out angles you can convert them to degrees first. Sin +ve (180 o – xo) Tan +ve All +ve (xo) Cos +ve (180 o + xo) (360 o – xo) = sin 150 o = sin(180 - 150) o = tan 300 o = -tan(360 - 300) = sin 30 o =- tan 60 o

Exercise 4 Find the exact value of: 05 -Jan-22 Exact Values