WAVE FUNCTIONS
Suppose we compare the following 3 graphs
(i)
y = 4sinx°
(ii)
y = 3cosx°
(iii)
y = 3cosx° + 4sinx°
(i)
y = 4sinx°
Main Features
(a) Wave shape
(b) Max = 4 when x = 90
(c) Min = -4 when x = 270
(ii)
y = 3cosx°
Main Features
(a) Roller-coaster shape
(b) Max = 3 when x = 0 or 360
(c) Min = -3 when x = 180
(iii)
y = 3cosx° + 4sinx°
Main Features
(a) Roller-coaster shape but to right of Y-axis  cos(x - …)
(b) Max = 5 when x  50
5cos(x – 50)°
(c) Min = -5 when x  230
(actual values are 53.1 & 233.1)
So
3cosx° + 4sinx° = 5cos(x – 53.1)°
By converting from a mixture of sinx & cosx to just cos(…..) or
possibly sin(…) the function is easier to deal with.
The function that uses just a single trig ratio – rather than a
mixture of sin & cos – is called a wave function.
We now look at why
3cosx° + 4sinx° = 5cos(x – 53.1)°
The Wave Function kcos(x - )°
Note:
any function in the form acosx° + bsinx°
kcos(x - )°
can be expressed in the form
where
a = kcos°
and
MUST
LEARN !!
b = ksin° .
By considering the trig addition formulae and comparing
coefficients we can prove the above as follows….
Proof
Let
acosx° + bsinx° = kcos(x - )°
= k(cosx°cos° + sinx°sin°)
= kcosx°cos° + ksinx°sin°
= (kcos°)cosx° + (ksin°)sinx°
Comparing coefficients we get
a = kcos° and
also
and
a2 + b2 =
=
=
=
b/
a
b = ksin°
(kcos°)2 + (ksin°)2
k2cos2° + k2sin2°
k2(cos2° + sin2°)
k2
= ksin° = tan°
kcos°
common factor!
cos2° + sin2° = 1
So
k2 = a2 + b2
and
tan = b/a
The Wave Function ksin(x + )°
Note:
any function in the form asinx° + bcosx°
ksin(x + )°
can be expressed in the form
where
a = kcos°
and
MUST
LEARN !!
b = ksin° .
By considering the trig addition formulae and comparing
coefficients we can prove the above as follows….
Proof
Let
asinx° + bcosx° = ksin(x + )°
= k(sinx°cos° + cosx°sin°)
= ksinx°cos° + kcosx°sin°
= (kcos°)sinx° + (ksin°)cosx°
Comparing coefficients we get
a = kcos° and
also
and
a2 + b2 =
=
=
=
b/
a
b = ksin°
(kcos°)2 + (ksin°)2
k2cos2° + k2sin2°
k2(cos2° + sin2°)
k2
= ksin° = tan°
kcos°
just like before
common factor!
cos2° + sin2° = 1
So
k2 = a2 + b2
and
tan = b/a
SUMMARY
acosx° + bsinx° = kcos(x - )°
asinx° + bcosx° = ksin(x + )°
In both cases..
a = kcos°
and
b = ksin°
and
tan ° = b/a
which leads to
k2 = a2 + b2