Trigonometry

1
Trigonometry
IB SL

• Common Identities
• Knowledge of double angle formulae
• Sin and Cos values you should know

2
Radians
y
As well as degrees, angles can be expressed in
RADIANS
?/2
1
?
0
x
RADIANS are the distance traveled around the unit
circle.
?
2?
3?/2
180o ? radians
360o 2? radians
0o 0 radians
90o ?/2 radians
270o 3 ?/2 radians
3
A proof
y
Using Pythagoras r12 r22 r2 (r cos ?) 2 (r
sin ?) 2 r2
?/2
r
r2
0
x
?
r2 cos2 ? r2 sin2 ? r2
r1
?
2?
(? r2)
cos2 ? sin2 ? 1
3?/2
4
Another proof
y
cos ? adj/hyp r1 / r
?/2
r1 r cos ?
sin ? opp/hyp r2 / r
r
r2
r2 r sin ?
x
0
?
?
r1
2?
tan ? opp/adj r2 / r1
tan ? r sin ? / r cos ?
3?/2
5
Double Angle Identities (1)
sin (2?)
A well know maths formula is ...
sin (A B) sin A cos B cos A sin B
Setting ? A B
sin (? ?) sin ? cos ? cos ? sin ?
sin (2?) 2 sin ? cos ?
6
Double Angles Identities (2)
cos (2?)
Another well know maths formula is ...
cos (A B) cos A cos B - sin A sin B
Setting ? A B
cos (? ?) cos ? cos ? - sin ? sin ?
cos (2?) cos2 ? - sin2 ?
Since, sin2 ? cos2 ? 1
cos (2?) cos2 ? (1 - cos2 ?) 2 cos2 ? - 1
cos (2?) 1 - sin2 ? sin2 ? 1 - 2 sin2 ?
7
Trigometrical Identities
Double Angle Formulae
These are all provided in your information
booklet to use
8
Things you should learn (1)
Isosceles triangle - short sides of 1
cos ? adj/hyp 1 / ?2
45
?2
sin ? opp/hyp 1 / ?2
1
45
sin 45 1/?2 0.707
1
cos 45 1/?2 0.707
Using Pythagoras h2 12 12 2 h ?2
9
Things you should learn (2)
Equilateral triangle - sides of 2
cos 60 adj/hyp 1 / 2
cos 30 adj/hyp ?3 / 2
2
2
sin 60 opp/hyp ?3 / 2
?3
sin 30 opp/hyp 1 / 2
60
60
Using Pythagoras 22 12 x2 x2 4 - 1 3 x
?3
10
Things you should learn (3)
cos 30 ?3 / 2
y sin x
sin 60 ?3 / 2
cos 60 1 / 2
sin 30 1 / 2
sin 0 0 sin 90 1
sin 45 1/?2
cos 45 1/?2
y cos x
sin x cos (90-x)
cos x sin (90-x)
cos 0 1 cos 90 0
11
Things you should know - summary
sin 0 0 sin 30 1/2 0.5 sin 45 1/?2
0.707 sin 60 ?3/2 0.866 sin 90 1
y sin x
cos 90 0 cos 60 1/2 0.5 cos 45 1/?2
0.707 cos 30 ?3/2 0.866 cos 0 1
y cos x
12
Have a go

• Page 254
• Identities
• Ex 8D
• Page 258
• Double Angles
• Ex 8E