Re: Trig Substitution

• To: mathgroup at smc.vnet.net
• Subject: [mg93461] Re: [mg93434] Trig Substitution
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Sun, 9 Nov 2008 05:25:59 -0500 (EST)

```x = Sin[t] + Sin[t]^2 + Sin[t]^3;

y = x /. {Sin[t_]^n_?EvenQ :>
(1 - Cos[t]^2)^(n/2),
Sin[t_]^n_ :>
(1 - Cos[t]^2)^((n - 1)/2) Sin[t]} // Expand

(-Sin[t])*Cos[t]^2 - Cos[t]^2 + 2*Sin[t] + 1

y == x // Simplify

True

Bob Hanlon

---- Bruce Colletti <bwcolletti at verizon.net> wrote:

=============
Re Mathematica 6.0.3 under WinXP.

Given a sum of (positive integer) powers of sin(t), I wish to cast (as much
as possible) the sum in terms of cos t, chiefly by substituting 1 - (cos
t)^2 for (sin t)^2 throughout.  The code below does this.

X=Sin[t]+Sin[t]^2+Sin[t]^3;

Y=Apply[List,X]/.{Sin[t]^n_?EvenQ:>Expand[(1-Cos[t]^2)^(n/2)],Sin[t]^n_:>Exp
and[(1-Cos[t]^2)^Quotient[n,2] Sin[t]]};

Apply[Plus,Y]

What is an easier way?  Thankx.

Bruce

--

Bob Hanlon

```

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