# Trigonometric rearrangement of tan(3x)cot(7x) to find limit

• Jan 27th 2013, 11:31 PM
mcleja
Trigonometric rearrangement of tan(3x)cot(7x) to find limit
I have tan(3x)cot(7x) and i need to get it in the form (sin(ax)/ax)/(sin(bx)/bx) for some a and b. I have use the tan(x)=sin(x)/cos(x) and cot(x)=cos(x)/sin(x) to get (sin(3x)/cos(3x))(cos(7x)/sin(7x)) and that is about as far as i got.
Could anybody help me to solve this rearrangement. Then I need to find the limit as x goes to zero. But i have to show it in the form stated above.

Thanks.
• Jan 28th 2013, 12:33 AM
mcleja
Re: Trigonometric rearrangement of tan(3x)cot(7x) to find limit
Could one use L'hopitals rule since (sin(3x)/cos(3x))(cos(7x)/sin(7x)) will equal 0/0. I tried this and got (sin(4x)(-7sin(7x))+4cos(7x)cos(4x))/(cos(4x)7cos(7x)+sin(7x)(-4sin(4x)) and i yo substitute in 0 you get 4/7 which i have confirmed is the right limit. But how do you go from my result to the form (sin(ax)/ax)/(sin(bx)/bx)?
• Jan 28th 2013, 07:46 AM
Soroban
Re: Trigonometric rearrangement of tan(3x)cot(7x) to find limit
Hello, mcleja!

You are expected to know these identities:

. . $\lim_{\theta\to0}\frac{\sin\theta}{\theta} \:=\:1\qquad \lim_{\theta\to0}\frac{\theta}{\sin\theta} \:=\:1$

Quote:

$\lim_{x\to0} \tan(3x)\cot(7x)$

We have: . $\tan(3x)\cot(7x) \;=\;\frac{\sin(3x)}{\cos(3x)}\cdot\frac{\cos(7x)} {\sin(7x)} \;=\;\frac{\sin(3x)}{1}\cdot \frac{1}{\sin(7x)}\cdot\frac{\cos(7x)}{\cos(3x)}$

. . $=\;\;{\color{red}\frac{3x}{3x}}\cdot\frac{\sin(3x) }{1}\cdot {\color{red}\frac{7x}{7x}}\cdot\frac{1}{\sin(7x)} \cdot\frac{\cos(7x)}{\cos(3x)} \;\;=\;\;\frac{3x}{7x}\cdot \frac{\sin(3x)}{3x}\cdot \frac{7x}{\sin(7x)}\cdot\frac{\cos(7x)}{\cos(3x)}$

. . $=\;\;\frac{3}{7}\cdot\frac{\sin(3x)}{3x}\cdot\frac {7x}{\sin(7x)}\cdot\frac{\cos(7x)}{\cos(3x)}$

Therefore: . $\lim_{x\to0}\left[ \frac{3}{7}\cdot\frac{\sin(3x)}{3x}\cdot\frac{7x}{ \sin(7x)}\cdot\frac{\cos(7x)}{\cos(3x)}\right] \;\;=\;\;\frac{3}{7}\cdot 1\cdot 1\cdot\frac{1}{1} \;\;=\;\;\frac{3}{7}$

• Jan 28th 2013, 01:11 PM
mcleja
Re: Trigonometric rearrangement of tan(3x)cot(7x) to find limit
ok i got that as the limit correctly but i could not get the expression tan(3x)cot(7x) in the form (sin(ax)/ax)/(sin(bx)/bx) which i need to show as an intermediate step for my assignment. Could you help me do that? Thank you for the reply!
• Jan 28th 2013, 03:42 PM
Prove It
Re: Trigonometric rearrangement of tan(3x)cot(7x) to find limit
You're not going to be able to get rid of the cosine terms, I expect that you are asked to have that form in your expression AMONG OTHER THINGS.

Anyway, notice that \displaystyle \begin{align*} \frac{\sin{(3x)}}{3x} \cdot \frac{7x}{\sin{(7x)}} \end{align*} is equivalent to \displaystyle \begin{align*} \frac{ \frac{\sin{(3x)}}{3x} }{ \frac{\sin{(7x)}}{7x} } \end{align*}.
• Jan 28th 2013, 07:44 PM
mcleja
Re: Trigonometric rearrangement of tan(3x)cot(7x) to find limit
Ok i see now. Thanks everybody.