Exact values

Exact Values

1. Block 3
Exact Angles

2. What is to be Learned?
•The exact trig values of vital angles
•How to use exact values with related
angles

3. Finding the angles
45
0
45
0
1
1
√2
H
O
A
sin45
0
=
1
/
√2
cos45
0
=
1
/
√2
tan45
0
=
1
/
1
= 1

4. Finding the angles
60
0
60
0
2
2
2
H
O
A
sin60
0
=
√3
/
2
cos60
0
=
1
/
2
tan60
0
=
√3
/
1
= √3
60
0
1 1
30
0
30
0
√3

5. Finding the angles
60
0
2
H
O
A
sin60
0
=
√3
/
2
cos60
0
=
1
/
2
tan60
0
=
√3
/
1
= √3
1
30
0
√3
cos30
0
=
√3
/
2
sin30
0
=
1
/
2
tan30
0
=
1
/
√3

6. And Finally
y = sinx y = cosx
Sin0
0
= 0
Sin90
0
= 1
Cos0
0
= 1
Cos90
0
= 0

7. Exact Values
There are some trig values we must
memorise – they are exact (not
approximations)

8. 00
00
30 30
00
4545
00
60 60
00
90 90
00
sin
cos
tan
0
π
/
2
π
/
3
π
/
4
π
/
6
degrees
rads
0 1
½
1
/
√2

√3
/
2
1
√3
/
2
1
/
√2
½ 0
0
1
/
√3 1√3 ∞

9. Finding all the relations
There are 4 related angles (usually)
Less than 90
0
Between 90
0
and 180
0
Between 180
0
and 270
0
Between 270
0
and 360
0
a
0
a
0
180 – a
180 + a
360 – a
320
0
140
0
220
0
40
0

10. Finding relations using NWD
a
0
180 – a
180 + a360 – a
iii
iii iv
Easiest when starting
in Quadrant 1
(Acute angle)
Relations of 70
0i 70
0
ii 180 – 70 = 110
0
iii 180 + 70 = 250
0
iv 360 – 70 = 290
0
AS
T C

11. Relations from non acute angles
Ex230
0
180 – a
180 + a
iii
iii iv
a
0
360 - a
AS
T C
Quad iii
so 180 + a = 230
0
a = 50
0
can easily find
rest of relations
180 – 50 = 130
0
360 – 50 = 310
0
0, 360
0
180
0
270
0
- need related acute angle
90
0

12. Relations from non acute angles
Ex330
0
180 – a
180 + a
iii
iii iv
a
0
360 – a
AS
T C
Quad iv
so 360 – a = 330
0
a = 30
0
rest of relations
180 – 30 = 150
0
180 + 30 = 210
0
0, 360
0
90
0
180
0
270
0
90
0

13. Find relations of these angles
1.100
0
80
0
260
0
280
0
2.200
0
20
0
160
0
340
0
3.300
0
60
0
120
0
240
0
4.195
0
15
0
165
0
345
0
5.308
0
52
0
128
0
232
0

14. Exact Values of non acute angles
Exsin 330
0
180 – a
180 + a
iii
iii iv
a
0
360 – a
AS
T C
Quad iv
so 360 – a
a = 30
0
sin 30
0
sin330
0
= ½
- need related acute angle
–
= 330
0 90
0
180
0
270
0
Acute Exact Value
= ½
Related Exact Value
Check Sign
0
0
,360
0

15. Exact Values of non acute angles
Extan 135
0
180 – a
180 + a
iii
iii iv
a
0
360 – a
AS
T C
Quad ii
so 180 – a
a = 45
0
tan 45
0
tan135
0
= 1
- need related acute angle
–
= 135
0 90
0
180
0
270
0
Acute Exact Value
= 1
Related Exact Value
Check Sign
0
0
,360
0

16. Exact Values of non acute angles
Excos 240
0
180 – a
180 + a
iii
iii iv
a
0
360 – a
AS
T C
Quad iii
so 180 + a
a = 60
0
cos 60
0
cos 240
0
= ½
- need related acute angle
–
= 240
0 90
0
180
0
270
0
0
0
,360
0
Acute Exact Value
= ½
Related Exact Value
Check Sign

17. You Try
1.sin120
0
2.cos240
0
3.tan135
0
4.sin225
0
5.tan210
0
6.cos330
0
7.sin180
0
(beware!)
√3
/
2
- ½
-1
-
1
/
√2
1
/
√3
√3
/
2
0
Key Question

18. It gets nastier!
Excos
5π
/
4
180 – a
180 + a
iii
iii iv
a
0
360 - a
AS
T C
Quad iii
so 180 + a = 225
0
a = 45
0
cos45
0
=
1
/
√2

cos225
0
= -
1
/
√2

55π
/
4
X
180
/
π
=
900 π
/
4 π
= 225
0
i.e. coscos
55π
/
4
=

-
1
/
√2