# Thread: Converting irrational number to surd form

1. ## Converting irrational number to surd form

I have the following function:

$\displaystyle y = \frac{cosx}{1 - sinx}$ where $\displaystyle x = \frac{\pi}{6}$

Say I solve this for y, and I get the solution 1.73205

I happen to know this is $\displaystyle \sqrt{3}$, but if I didn't, is there an intuitive way to make this connection? It seems to be bad form to express it as a decimal.

Thanks.

2. Hello, Glitch!

I have the following function:
. .
$\displaystyle y \:=\: \dfrac{\cos x}{1 - \sin x}$ where $\displaystyle x = \frac{\pi}{6}$

Say I solve this for $\displaystyle y$, and I get the solution 1.73205

I happen to know this is $\displaystyle \sqrt{3}$,
but if I didn't, is there an intuitive way to make this connection?
It seems to be bad form to express it as a decimal.

We're expected to know about the 30-60 right triangle.

Code:
                              *
* 60*
2     *       *
*           * 1
*               *
* 30                *
*   *   *   * _ *   *   *
√3

Hence, we have: .$\displaystyle \begin{Bmatrix} \sin30^o &=& \sin\frac{\pi}{6} &=& \dfrac{1}{2} \\ \\[-3mm] \cos30^o &=& \cos\frac{\pi}{6} &=&\dfrac{\sqrt{3}}{2} \\ \\[-3mm] \tan30^o &=& \tan\frac{\pi}{6} &=& \dfrac{1}{\sqrt{3}} \end{Bmatrix}$

So we have: .$\displaystyle y \;=\;\dfrac{\cos\frac{\pi}{6}}{1-\sin\frac{\pi}{6}} \;=\;\dfrac{\frac{\sqrt{3}}{2}}{1 - \frac{1}{2}} \;=\;\dfrac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} \;=\;\sqrt{3}$

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# how to convert decimals to surds and back to decimals

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