Using Maple, how can I rewrite a trigonometric expression in phase-amplitude form?
To rewrite a trigonometric expression in phase-amplitude form in Maple, consider the following example. Suppose you want to rewrite the following expression:
a*cos(x) + b*sin(x)You wish it to be an expression of the form:
C*cos(x + omega)Although there is no built-in function for it, you can do this manually, as follows:
> ex := a*cos(x) + b*sin(x); ex := a cos(x) + b sin(x)First, expand the target expression, for comparison:
> expand( A*cos(x+B) ); A cos(x) cos(B) - A sin(x) sin(B)From this expression, you can see how the terms match up. If you make the appropriate substitution, you can combine the terms using trigonometric identities, and then substitute back the original variables:
> subs(a=A*cos(B),b=A*sin(B),ex); A cos(x) cos(B) + A sin(x) sin(B) > combine(%, trig); A cos(x - B) > subs(A=a/cos(B), B=arctan(b/a), %); a cos(x - arctan(b/a)) ---------------------- cos(arctan(b/a)) > simplify(%); / 2 2\l/2 |a + b | a |-------| cos(x - arctan(b/a)) | 2 | \ a / > simplify(%,symbolic); 2 2 1/2 (a + b ) cos(x - arctan(b/a))
The symbolic option above does "naive" simplification,
basically treating all variables as real and positive.
For more about Maple, see Maple at IU.
For more about statistical and mathematical software, email the UITS Stat/Math Center, visit the center's web page, or phone 812-855-4724 (IUB) or 317-278-4740 (IUPUI). The center is located in Bloomington at 410 N. Park Avenue, and is open for consultation by appointment Monday-Friday 9am-5pm.
Last modified on January 28, 2011.
