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.

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)?

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

Hello, mcleja!

You are expected to know these identities:

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

Quote:

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

We have: .$\displaystyle \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)} $

. . $\displaystyle =\;\;{\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)} $

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

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

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!

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 $\displaystyle \displaystyle \begin{align*} \frac{\sin{(3x)}}{3x} \cdot \frac{7x}{\sin{(7x)}} \end{align*}$ is equivalent to $\displaystyle \displaystyle \begin{align*} \frac{ \frac{\sin{(3x)}}{3x} }{ \frac{\sin{(7x)}}{7x} } \end{align*}$.

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

Ok i see now. Thanks everybody.