Question

write sin 2x in terms of t = tanx

Answer 1

hint: sin 2x = 2 sin (x) cos (x)

Answer 2

yes i know that. can you walk me through the steps to complete it?

Answer 3

This is essentially the Weierstrass substitution.

http://en.wikipedia.org/wiki/Weierstrass_substitution

Answer 4

http://en.wikipedia.org/wiki/Weierstrass_substitution#Derivation

Answer 5

I want to know how they subbed x in for x/2 when t = tan x/2 ...hm

Answer 6

can someone please walk me through the steps please?

Answer 7

Pretend \(x\) is \(2x\) and then the \(\tan\frac{x}2\) is just \(\tan x\).

Answer 8

$$\begin{align*}\sin2x&=2\sin x\cos x\text{ by the double-angle identity}\\&=2\tan x\cos^2 x\text{ by }\tan x=\frac{\sin x}{\cos x}\\&=\frac{2\tan x}{\sec^2 x}\text{ by }\sec x=\frac1{\cos x}\\&=\frac{2\tan x}{1+\tan^2 x}\text{ by }\sec^2x=1+\tan^2 x\\&=\frac{2t}{1+t^2}\text{ by }t=\tan^2 x\end{align*}$$

Answer 9

\(t=\tan x\) oops

Answer 10

wait wait, step 2 over again? :( where tan x = sin (x)/cos (x) identity part?

Answer 11

$$2\sin x\cos x=2(\tan x\cos x)\cos x=2\tan x\cos^2 x$$

Answer 12

why does sinx = tanxcosx?

Answer 13

sin = tan (x) * cos (x)

Answer 14

OH I SEE.

Answer 15

ok i get it now :)

Answer 16

so instead of replacing tan^2 (x) + 1 with sec^2 (x) you subbed t = tan x for tan^2 (x)?

Answer 17

Ohh because you want it in terms of t =tan (x)

Answer 18

Yep :-p