445.102 Mathematics 2
Module 4
Cyclic Functions
Lecture 4
Compounding the Problem
Angle Formulae
In this lecture we treat sine, cosine & tangent
as mathematical functions which have
relationships with each other. These are
expressed as various formulae.
It is important that you UNDERSTAND this
work, but not that you can reproduce it. We
would like you to be able to USE the formulae
when needed. We want you to become
familiar with using cyclic functions in
algebraic expressions.
f(x)
= sin x
g(x) = A + sin x
Vertical shift of A
h(x) = sin(x + A) Horizontal shift of –A
j(x) = sin (Ax)
Horizontal squish A times
k(x) = Asin x
Vertical stretch A times
m(x) = n(x) sin x Outline shape n(x)
Post-Lecture Exercise
f(x) = sin (–x)
1.0 0
1.0 0
2.0 0
3.0 0
4.0 0
5.0 0
6.0 0
7.0 0
1.0 0
2.0 0
3.0 0
4.0 0
5.0 0
6.0 0
7.0 0
-1.0 0
f(x) = cos (–x)
1.0 0
-1.0 0
Post-Lecture Exercise
f(x) = 2cos (x/2)
f(x) = 3sin (2x)
5.0
5.0
š
2š
š
-5.0
-5.0
f(x) = 2 + sin(x/3)
5.0
š
-5.0
2š
2š
Post-Lecture Exercise
3.
T(t) = 38.6 + 3sin(πt/8)
a) 38.6 is the normal temperature
b) 38.6 + 3sin(πt/8) = 40
<=> 3sin(πt/8) = 1.4
<=> sin(πt/8) = 1.4/3 = 0.467
<=> πt/8 = sin-1(0.467) = 0.486
<=> t = 0.486*8/π = 1.236
after about 1 and a quarter days.
4.
Maximum is where sine is minimum
i.e. when D = 8 + 2 = 10metres
445.102
Lecture 4/4
Administration
Last
Lecture
Distributive
Functions
Compound Angle
Formulae
Double Angle Formulae
Sum and Product Formulae
Summary
The Distributive Law





2(a + b) = 2a + 2b
(a + b)2 ≠ a2 + b2
= a2 + 2ab + b2
(a + b)/2 = a/2 + b/2
log(a + b) ≠ log a + log b
= log a . log b
sin (a + b) ≠ sin a + sin b
= ????????????
The Unit Circle Again
sin b
b a
sin (a + b) < sin a + sin b
sin a
A Graphical Explanation
1.0 0
0.5 0
-0.5 0
-1.0 0
sin (a+b)
sin b
sin a
a b (a+b)
š
445.102
Lecture 4/4
Administration
Last
Lecture
Distributive Functions
Compound Angle
Double Angle
Formulae
Formulae
Sum & Product Formulae
Summary
The Formula for 0 ≤ ø ≤ π/2
x
b a
y
sin b
z
sin a
Lecture 4/5 – Summary
Compound Angle Formulae






sin (A + B) = sinA.cosB + cosA.sinB
sin (A – B) = sinA.cosB – cosA.sinB
cos (A + B) = cosA.cosB – sinA.sinB
cos (A – B) = cosA.cosB + sinA.sinB
tan (A + B) = (tanA + tanB)
1 – tanA.tanB
tan (A – B) = (tanA – tanB)
1 + tanA.tanB
Shelter from the Storm
7m
4m
ø
4 cosø + 7sinø
Shelter from the Storm
7m
4m
ø
4 cosø + 7sinø
√65
µ
7
4
Shelter from the Storm
7m
4m
ø
4 cosø + 7sinø
√65
µ
7
4
sinµ = 4/√65
4 = √65 sinµ
cosµ = 7/√65
7 = √65 cosµ
Shelter from the Storm
7m
4m
ø
√65sinµ cosø + √65cosµsinø
√65
µ
7
4
sinµ = 4/√65
4 = √65 sinµ
cosµ = 7/√65
7 = √65 cosµ
445.102
Lecture 4/4
Administration
Last
Lecture
Distributive Functions
Compound Angle Formulae
Double Angle
Sum
Formula
& Product Formulae
Summary
Double Angle Formulae






sin (A + B) = sinA.cosB + cosA.sinB
sin 2A = sinA.cosA + cosA.sinA
= 2sinA cosA
cos (A + B) = cosA.cosB – sinA.sinB
cos 2A = cosA.cosA – sinA.sinA
= cos2A – sin2A
Double Angle Formulae

tan (A + B) = (tanA + tanB)
1 – tanA.tanB

tan 2A = (tanA + tanA)
1 – tanA.tanA

tan 2A =
2tanA
1 – tan2A
445.102
Lecture 4/4
Administration
Last
Lecture
Distributive Functions
Compound Angle Formulae
Double Angle Formula
Sum
& Product Formulae
Summary
The Octopus
Large wheel, radius 6m, 8 second period.
A = 6sin(2πx/8)
10 .0
5.0
2.00
-5.0
-10 .0
4.00
6.00
8.00
10 .00
12 .00
14 .00
16 .00
The Octopus
Add a small wheel, radius 1.5m, 2s period.
B = 1.5sin(2πx/2)
10 .0
5.0
2.00
-5.0
-10 .0
4.00
6.00
8.00
10 .00
12 .00
14 .00
16 .00
The Octopus
Combine the two......
A + B = 6sin(2πx/8) + 1.5sin(2πx/2)
10 .0
5.0
2.00
-5.0
-10 .0
4.00
6.00
8.00
10 .00
12 .00
14 .00
16 .00
The Surf
Decent surf has a height of 1.5m, 15s period.
A = 1.5sin(2πx/15)
2.00
1.00
5.0
-1.00
-2.00
10 .0
15 .0
20 .0
25 .0
30 .0
35 .0
40 .0
The Surf
Add similar wave, say: 1m, 13s period.
A + B = 1.5sin(2πx/15) + 1sin(2πx/13)
4.00
2.00
-20 0
-15 0
-10 0
-50
50
-2.00
-4.00
10 0
15 0
20 0
25 0
Adding Sine Functions






sin(A+B) = sinAcosB + sinBcosA
sin(A–B) = sinAcosB – sinBcosA
Adding.........
sin(A+B) + sin(A–B) = 2sinAcosB
Rearranging.........
sinAcosB = 1/2[sin(A+B) + sin(A–B)]
Adding Sine Functions

sinAcosB = 1/2[sin(A+B) + sin(A–B)]
Or, making A = (P+Q)/2 and B = (P–Q)/2
That is: A+B = 2P/2 and A–B = 2Q/2
1/ [sin P + sin Q] = sin (P+Q)/ cos (P–Q)/
2
2
2

sin P + sin Q = 2 sin (P+Q)/2 cos (P–Q)/2



445.102
Lecture 4/4
Administration
Last
Lecture
Distributive Functions
Explanations of sin(A + B)
Developing a Formula
Further Formulae
Summary
Lecture 4/4 – Summary
Compounding the Problem



Please KNOW THAT these
formulae exist
Please BE ABLE to follow the
logic of their derivation and use
Please PRACTISE the simple
applications of the formulae as in
the Post-Lecture exercises