anonymous
  • anonymous
write sin 2x in terms of t = tanx
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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Jhannybean
  • Jhannybean
hint: sin 2x = 2 sin (x) cos (x)
anonymous
  • anonymous
yes i know that. can you walk me through the steps to complete it?
anonymous
  • anonymous
This is essentially the Weierstrass substitution. http://en.wikipedia.org/wiki/Weierstrass_substitution

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anonymous
  • anonymous
http://en.wikipedia.org/wiki/Weierstrass_substitution#Derivation
Jhannybean
  • Jhannybean
I want to know how they subbed x in for x/2 when t = tan x/2 ...hm
anonymous
  • anonymous
can someone please walk me through the steps please?
anonymous
  • anonymous
Pretend \(x\) is \(2x\) and then the \(\tan\frac{x}2\) is just \(\tan x\).
anonymous
  • anonymous
$$\begin{align*}\sin2x&=2\sin x\cos x\text{ by the double-angle identity}\\&=2\tan x\cos^2 x\text{ by }\tan x=\frac{\sin x}{\cos x}\\&=\frac{2\tan x}{\sec^2 x}\text{ by }\sec x=\frac1{\cos x}\\&=\frac{2\tan x}{1+\tan^2 x}\text{ by }\sec^2x=1+\tan^2 x\\&=\frac{2t}{1+t^2}\text{ by }t=\tan^2 x\end{align*}$$
anonymous
  • anonymous
\(t=\tan x\) oops
Jhannybean
  • Jhannybean
wait wait, step 2 over again? :( where tan x = sin (x)/cos (x) identity part?
anonymous
  • anonymous
$$2\sin x\cos x=2(\tan x\cos x)\cos x=2\tan x\cos^2 x$$
anonymous
  • anonymous
why does sinx = tanxcosx?
Jhannybean
  • Jhannybean
sin = tan (x) * cos (x)
Jhannybean
  • Jhannybean
OH I SEE.
anonymous
  • anonymous
ok i get it now :)
Jhannybean
  • Jhannybean
so instead of replacing tan^2 (x) + 1 with sec^2 (x) you subbed t = tan x for tan^2 (x)?
Jhannybean
  • Jhannybean
Ohh because you want it in terms of t =tan (x)
anonymous
  • anonymous
Yep :-p

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