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[MUG] Combining trigonometrical expressions
| [MUG] Combining trigonometrical expressions |
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Author: Michael Brennan
Posted: 03/03/2000 13:48:33 GMT
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>> From: (Michael Brennan)
Hi Muggers,
The symmetry from the binomial theorem suggests that cos(17x) and
sin(17x) are both described by identical polynomials albeit that the
cos(x) and sin(x) are switched in these polynomials.
Below I have tried to show that once expanded they are coincident when
considered as functions of cos(x) and sin(x) alone i.e the generating
function would be the same.
The problem is that while I get an expression for the cos(17x) in terms
of cos(x), the sin(17x) has mixed terms ...how can I get the sin(17x) in
terms of just sin(x). TIA.
Michael.
> eq1:=expand((cos(x)+I*sin(x))^17);
17 16 2 15
eq1 := cos(x) + 17 I sin(x) cos(x) - 136 sin(x) cos(x)
3 14 4 13
- 680 I sin(x) cos(x) + 2380 sin(x) cos(x)
5 12 6 11
+ 6188 I sin(x) cos(x) - 12376 sin(x) cos(x)
7 10 8 9
- 19448 I sin(x) cos(x) + 24310 sin(x) cos(x)
9 8 10 7
+ 24310 I sin(x) cos(x) - 19448 sin(x) cos(x)
11 6 12 5
- 12376 I sin(x) cos(x) + 6188 sin(x) cos(x)
13 4 14 3
+ 2380 I sin(x) cos(x) - 680 sin(x) cos(x)
15 2 16 17
- 136 I sin(x) cos(x) + 17 sin(x) cos(x) + I sin(x)
> a:=evalc(Re(eq1));
17 2 15 4 13
a := cos(x) - 136 sin(x) cos(x) + 2380 sin(x) cos(x)
6 11 8 9
- 12376 sin(x) cos(x) + 24310 sin(x) cos(x)
10 7 12 5
- 19448 sin(x) cos(x) + 6188 sin(x) cos(x)
14 3 16
- 680 sin(x) cos(x) + 17 sin(x) cos(x)
> simplify(a);
17 15 13
65536 cos(x) - 278528 cos(x) + 487424 cos(x)
11 9 7
- 452608 cos(x) + 239360 cos(x) - 71808 cos(x)
5 3
+ 11424 cos(x) - 816 cos(x) + 17 cos(x)
> b:=evalc(Im(eq1));
16 3 14
b := 17 sin(x) cos(x) - 680 sin(x) cos(x)
5 12 7 10
+ 6188 sin(x) cos(x) - 19448 sin(x) cos(x)
9 8 11 6
+ 24310 sin(x) cos(x) - 12376 sin(x) cos(x)
13 4 15 2 17
+ 2380 sin(x) cos(x) - 136 sin(x) cos(x) + sin(x)
> simplify(b);
16 14
65536 sin(x) cos(x) - 245760 sin(x) cos(x)
12 10
+ 372736 sin(x) cos(x) - 292864 sin(x) cos(x)
8 6
+ 126720 sin(x) cos(x) - 29568 sin(x) cos(x)
4 2
+ 3360 sin(x) cos(x) - 144 sin(x) cos(x) + sin(x)
>
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| [MUG] Re: Combining trigonometrical expressions |
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Author: Maple Group
Posted: 09/03/2000 13:41:16 GMT
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Subject: Combining trigonometrical expressions
The following Maple session shows how you can control the simplification
process in this type of case.
---------------------------------------------------------------------------
|\^/| Maple V Release 5.1 (University of Waterloo)
._|\| |/|_. Copyright (c) 1981-1998 by Waterloo Maple Inc. All rights
\ MAPLE / reserved. Maple and Maple V are registered trademarks of
<____ ____> Waterloo Maple Inc.
| Type ? for help.
#
# |
| [MUG] Re: Combining trigonometrical expressions |
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Author: Michael Monagan
Posted: 10/03/2000 08:05:55 GMT
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>> From: Michael Monagan
Much simpler is
algsubs( cos(x)^2=1-sin(x)^2, expression );
Mike
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