Exact values
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Transcript of 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
- need related acute angle
–
= 3300 900
1800
2700
Acute Exact Value
= ½ Related Exact Value
Check Sign
00,3600
Exact Values of non acute angles
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
- need related acute angle
–
= 1350 900
1800
2700
Acute Exact Value
= 1 Related Exact Value
Check Sign
00,3600
Exact Values of non acute angles
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
- need related acute angle
–
= 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