Solving the Sin(theta)/theta Equation: A Puzzling Problem

[brycenrg]

Homework Statement

Homework Equations

How does he get sin(theta)/theta / 1+ sintheta/theta *1/costheta
I don't understand how he goes from sinetheta/theta + tantheta to sin(theta)/theta / 1+ sintheta/theta *1/costheta

to me it looks like he just brought up the theta like theta/1 and just bumped up the theta but i know you can't do that.

The Attempt at a Solution

Ive tried too many times, there is my question if anyone can help. I appreciate it.

 

[scurty]

1/theta was multiplied in both the numerator and denominator (i.e. Multiplying by 1). It's a type of algebraic manipulation trick, similar to cleverly adding 0 to an expression to simplify it.

 

[PeroK]

Another method: you could try inverting the original limit and see what you get:

$$\lim_{θ→0} \frac{θ+tanθ}{sinθ}$$

 

[brycenrg]

Thank you, I remember that trick. I don't see it though because if you times it by theta/theta wouldn't it be thetasin(theta)/ 2theta + tantheta? Even if i did (1/theta) / (1/theta) its seems not to work because 1/costheta would be 1/theta(cos(theta))..dang i don't see it.

 

[brycenrg]

So i see that it would work for (1/theta)(1/theta) is that what you mean? i mean that is the same thing as theta over theta but you have to put it in that form for it to work

 

[scurty]

So i see that it would work for (1/theta)(1/theta) is that what you mean? i mean that is the same thing as theta over theta but you have to put it in that form for it to work

Yeah, that's what I mean. The whole point of doing that is so you would have a $\frac{sin(\theta)}{\theta}$ in the expression, which you know the limit of.