Compound angle formulae intro

Compound Angle Formulae

1. Block 2
Compound Angle Formulae
Introduction

2. What is to be learned?
•How to “break brackets” with trig stuff.
•How to be cunning as a fox

3. 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 ∞
Need to Know!

4. Breaking Brackets Trig Style
Cos(A + B)
= CosA + CosB ??
Cos(30
0
+ 60
0
)
= Cos30
0
+ Cos60
0
= √3
/
2
+
1
/
2

= Cos(90= Cos(90
00
))= 0= 0

5. In Fact
Cos(A + B) =
Cos(30
0
+ 60
0
) =
Cos 90
0
CosA CosBCosA CosB– SinA SinBSinA SinB
Cos30Cos30
00
Cos60 Cos60
00
– Sin30Sin30
00
Sin60 Sin60
00
= =
√3√3
//
22 X X
11
//
22 –
11
//
22 X X
√3√3
//
22
= =
√3√3
//
44 –
√3√3
//
44
= 0= 0

6. Also
Cos(A – B) =
Cos(A + B) =
Cos(A B) =
CosA CosBCosA CosB+ SinA SinB+ SinA SinB
CosA CosBCosA CosB– SinA SinBSinA SinB
CosA CosBCosA CosB SinA SinBSinA SinB++
– ++
–

7. Very Common Question
Cos P =
5
/
13
Sin Q =
1
/
√ 5
Cos(P – Q)?
=CosP CosQ + SinP SinQ
Must be cunning
PP
OO
HH
AA
AA
HH
55
1313
??
??
22
= 13 = 13
22
– 5 5
22
??
22
= 144 = 144
? = 12? = 12
1212
Sin P = Sin P =
1212
//
1313
√√ √√???? √√

8. Very Common Question
Cos P =
5
/
13
Sin Q =
1
/
√ 5
Cos(P – Q)?
=CosP CosQ + SinP SinQ
Must be cunning
PP
OO
HH
AA
55
1313
??
22
= ( = (√5)
22
– 1 1
22
??
22
= 5 = 5 – 1 1
??
22
= 4 = 4
1212
Cos Q = Cos Q =
22
//
√5√5
√√ √√√√
Q
1
√5
? = 2
2
√√
Sin P = Sin P =
1212
//
1313
??

9. Very Common Question
Cos P =
5
/
13
Sin Q =
1
/
√ 5
Cos(P – Q)?
=CosP CosQ + SinP SinQ
=
5
/
13
X
2
/
√5
+
12
/
13
X
1
/
√5
= 10 + 12
Cos Q = Cos Q =
22
//
√5√5
√√ √√√√√√
Sin P = Sin P =
1212
//
1313
13√5 13√5
= 22
13√5

10. Compound Angle Formulae
Cos(A – B) =
Cos(A + B) =
On Formula Sheet as
Cos(A B) =
CosA CosBCosA CosB+ SinA SinB+ SinA SinB
CosA CosBCosA CosB– SinA SinBSinA SinB
CosA CosBCosA CosB SinA SinBSinA SinB
++
– ++
–

11. Very Common Question
Cos P =
15
/
17
Sin Q =
2
/
√ 13
Cos(P – Q)?
=CosP CosQ + SinP SinQ
PP
OO
HH
AA
AA
HH
1515
1717
??
??
22
= 17 = 17
22
– 15 15
22
??
22
= 64 = 64
? = 8? = 8
88
Sin P = Sin P =
88
//
1717
√√ √√???? √√

12. Very Common Question
Cos P =
15
/
17
Sin Q =
2
/
√ 13
Cos(P – Q)?
=CosP CosQ + SinP SinQ
PP
OO
HH
AA
55
1313
??
22
= ( = (√13)
22
– 2 2
22
??
22
= 13 = 13 – 44
??
22
= 9 = 9
1212
Cos Q = Cos Q =
33
//
√13√13
√√ √√√√
Q
2
√13
? = 3
3
√√
Sin P = Sin P =
88
//
1717
??

13. Very Common Question
Cos P =
15
/
17
Sin Q =
2
/
√ 13
Cos(P – Q)?
=CosP CosQ + SinP SinQ
=
15
/
17
X
3
/
√13
+
8
/
17
X
2
/
√13
= 45 + 16
Cos Q = Cos Q =
33
//
√13√13
√√ √√√√√√
Sin P = Sin P =
88
//
1717
17√13 17√13
= 61
17√13

14. Sin A =
4
/
5
, Cos B =
3
/
√10
Find the value

of Cos (A + B)
Using triangles Cos A =
3
/
5
, Sin B =
1
/
√10
Cos (A + B) = CosA CosB – SinA SinB
=
3
/
5
X
3
/
√10
–
4
/
5
X
1
/
√10
=
9
/
5√10
–
4
/
5√10
=
5
/
5√10
=
1
/
√10
Key Question

15. Other Common Type Question
Exact value of cos 75
0
?
Angles that we do know
0 30 45 60 90
But not 75
cos75
0
=cos(30
0
+ 45
0
)
=cos 30
0
cos 45
0
– sin 30
0
sin 45
0
=
+
√3
/
2
X
1
/
√2– ½ X
1
/
√2
= √3
2√2
– 1
2√2
= √3 – 1
2√2

16. Other Common Type Question
Exact value of cos 15
0
?
Angles that we do know
0 30 45 60 90
But not 15
cos15
0
=cos(60
0
– 45
0
)
=cos 60
0
cos 45
0
+ sin 60
0
sin 45
0
=
1
/
2
X
1
/
√2+
√3
/
2
X
1
/
√2
= 1
2√2
+ √3
2√2
= 1 + √3
2√2
Usually need
to rationalise
denominator

17. Exact value of cos 105
0
?
=cos(60
0
+ 45
0
)
=cos 60
0
cos 45
0
– sin 60
0
sin 45
0
= –
√3
/
2
X
1
/
√2
½ X
1
/
√2
= 1
2√2
– √3
2√2
= 1 – √3
2√2
Key Question 2

18. Working Backwards
Cos(A – B) =CosA CosBCosA CosB+ SinA SinB+ SinA SinB
Cos(R – P) =CosR CosPCosR CosP+ SinR SinP+ SinR SinP
CosM CosGCosM CosG+ SinM SinG+ SinM SinGCos(M – G) =
Cos70Cos70
00
Cos10 Cos10
00
+ Sin70+ Sin70
00
Sin10 Sin10
00
Cos(70 – 10) =
= Cos60
0

= ½