exact values

Block 3

Exact Angles

What is to be Learned?

• The exact trig values of vital angles• How to use exact values with related

angles

Finding the angles

450

450

1

1

√2H

O

A

sin450 = 1/√2

cos450 = 1/√2

tan450 = 1/1 = 1

Finding the angles

600600

2

2

2H

O

A

sin600 = √3/2

cos600 = 1/2

tan600 = √3/1 = √3600

1 1

300 300

√3

Finding the angles

600

2H

O

A

sin600 = √3/2

cos600 = 1/2

tan600 = √3/1 = √3

1

300

√3

cos300 = √3/2

sin300 = 1/2

tan300 = 1/√3

And Finallyy = sinx y = cosx

Sin00 = 0

Sin900 = 1

Cos00 = 1

Cos900 = 0

Exact Values

There are some trig values we must memorise – they are exact (not approximations)

0000 30 3000 454500 60 6000 90 9000

sin

cos

tan

0 π/2π/3

π/4π/6

degrees

rads

0 1½ 1/√2 √3/2

1 √3/21/√2 ½ 0

0 1/√3 1 √3 ∞

Finding all the relationsThere are 4 related angles (usually)

Less than 900Between 900 and 1800

Between 1800 and 2700 Between 2700 and 3600

a0

a0

180 – a

180 + a 360 – a 3200

1400

2200

400

Finding relations using NWD

a0180 – a

180 + a 360 – a

iii

iii iv

Easiest when starting

in Quadrant 1

(Acute angle)

Relations of 700 i 700

ii 180 – 70 = 1100

iii 180 + 70 = 2500

iv 360 – 70 = 2900

AS

T C

Relations from non acute angles

Ex 2300

180 – a

180 + a

iii

iii iv

a0

360 - a

AS

T C

Quad iiiso 180 + a = 2300

a = 500

can easily find rest of relations

180 – 50 = 1300

360 – 50 = 3100

0, 36001800

2700

- need related acute angle

900

Relations from non acute angles

Ex 3300

180 – a

180 + a

iii

iii iv

a0

360 – a

AS

T C

Quad ivso 360 – a = 3300

a = 300

rest of relations

180 – 30 = 1500

180 + 30 = 2100

0, 3600

900

1800

2700

900

Find relations of these angles1. 1000

800 2600 2800

2. 2000

200 1600 3400

3. 3000

600 1200 2400

4. 1950

150 1650 3450

5. 3080

520 1280 2320

Exact Values of non acute angles

Ex sin 3300

180 – a

180 + a

iii

iii iv

a0

360 – a

AS

T C

Quad ivso 360 – a

a = 300

sin 300

sin3300 = ½


–

= 3300 900

1800

2700

Acute Exact Value

= ½ Related Exact Value

Check Sign

00,3600


Ex tan 1350

180 – a

180 + a

iii

iii iv

a0

360 – a

AS

T C

Quad iiso 180 – a

a = 450

tan 450

tan1350 = 1


–

= 1350 900

1800

2700

Acute Exact Value

= 1 Related Exact Value

Check Sign

00,3600


Ex cos 2400

180 – a

180 + a

iii

iii iv

a0

360 – a

AS

T C

Quad iiiso 180 + a

a = 600

cos 600

cos 2400 = ½


–

= 2400 900

1800

2700

00,3600

Acute Exact Value

= ½ Related Exact Value

Check Sign

You Try

1. sin1200

2. cos2400

3. tan1350

4. sin2250

5. tan2100

6. cos3300

7. sin1800 (beware!)

√3/2

- ½ -1

- 1/√2

1/√3√3/2

0

Key Question

It gets nastier!

Ex cos5π/4

180 – a

180 + a

iii

iii iv

a0

360 - a

AS

T C

Quad iiiso 180 + a = 2250

a = 450

cos450 = 1/√2

cos2250 = - 1/√2

55π/4 X 180/ π = 900 π/4 π

= 2250

i.e. coscos55π/4 = - 1/√2

exact values

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