The value of 24 is:

Question

The value of $\cot\dfrac\pi{24}$ is:

Accepted Answer

$$\cot \theta  = {{1 + \cos 2\theta } \over {\sin 2\theta }} = {{1 + \left( {{{\sqrt 3  + 1} \over {2\sqrt 2 }}} \right)} \over {\left( {{{\sqrt 3  - 1} \over {2\sqrt 2 }}} \right)}}$$

$$\theta  = {\pi  \over {24}}$$

$$ \Rightarrow \cot \left( {{\pi  \over {24}}} \right) = {{1 + \left( {{{\sqrt 3  + 1} \over {2\sqrt 2 }}} \right)} \over {\left( {{{\sqrt 3  - 1} \over {2\sqrt 2 }}} \right)}}$$

$$ = {{\left( {2\sqrt 2  + \sqrt 3  + 1} \right)} \over {\left( {\sqrt 3  - 1} \right)}} \times {{\left( {\sqrt 3  + 1} \right)} \over {\left( {\sqrt 3  + 1} \right)}}$$

$$ = {{2\sqrt 6  + 2\sqrt 2  + 3 + \sqrt 3  + \sqrt 3  + 1} \over 2}$$

$$ = \sqrt 6  + \sqrt 2  + \sqrt 3  + 2$$

Solutions · 1