Orthogonality of Sines and Cosines

​
sine-sine
cosine-cosine
sine-cosine
m
1
n
1
This example comes from[1], Section 9.2, Modeling with First Order Equations.
This Demonstration shows the orthogonality of sines and cosines.
For any
m
and
n
,
L
∫
-L
sin
mπx
L
cos
nπx
L
dx=0
.
For
m≠n
,
L
∫
-L
sin
mπx
L
sin
nπx
L
dx=0
,
L
∫
-L
cos
mπx
L
cos
nπx
L
dx=0
.
For
m=n
,
L
∫
-L
2
sin
mπx
L
dx=L
,
L
∫
-L
2
cos
mπx
L
dx=L
.
As you change
m
and
n
, you can see graphically that the areas above and below the axis cancel out in the first three cases. The orthogonal properties of sine and cosine are used in the solution of some classes of partial differential equations in terms of Fourier series.

References

[1] J. R. Brannan and W. E. Boyce, Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications, New York: John Wiley and Sons, 2010.
​

External Links

Differential Equation (Wolfram MathWorld)
Orthogonal Functions (Wolfram MathWorld)
Fourier Series (Wolfram MathWorld)

Permanent Citation

Stephen Wilkerson
​
​"Orthogonality of Sines and Cosines"​
​http://demonstrations.wolfram.com/OrthogonalityOfSinesAndCosines/​
​Wolfram Demonstrations Project​
​Published: December 7, 2010