WAVE FUNCTIONS

1
Maxima Minima
We have already seen that 3sinx 4cosx
5cos(x 53.1)
It now follows that
the maximum value of 3sinx 4cosx is
5
which occurs when angle 0
ie x 53.1 0 or x 53.1
the minimum value of 3sinx 4cosx is
-5
which occurs when angle 180
ie x 53.1 180 or x 233.1
Also
3sinx 4cosx 8 5cos(x 53.1) 8
So the maximum value of 3sinx 4cosx 8
is
5 8 or 13
the minimum value of 3sinx 4cosx 8
is
-5 8 or 3
2
Example
Express 7cosx 24sinx in the form kcos(x
- ?) .
Hence find the maximum value of
1 7cosx 24sinx 30
and the corresponding value of x.

a 7 b 24
kcos?gt0 Q1 or Q4
kcos? 7 and ksin? 24
ksin?gt0 Q1 or Q2
(kcos?)2 (ksin?)2 72 242
k2 49 576
?
?
?
k2 625
?
k 25
? in Q1
tan? b/a
24/7
So 7cosx 24sinx 25cos(x 73.7)
tan-1(24/7) 73.7
Minimum of 7cosx 24sinx
minimum of 25cos(x 73.7)
-25
Maximum of fraction
1 -25 30
1/5
when x 73.7 180
ie x 253.7