Blog: `6(cos((5pi)/12) + i sin((5 pi)/12))` Write the standard form of the complex number.

Wednesday, March 31, 2010

$\displaystyle{6}{\left({\cos{{\left(\frac{{{5}\pi}}{{12}}\right)}}}+{i}{\sin{{\left(\frac{{{5}\pi}}{{12}}\right)}}}\right)}$ Write the standard form of the complex number.

You need to go from trigonometric form in standard form of the complex number, hence, you just need to simplify it replacing the values of trigonometric functions for $\displaystyle{\cos{{\left({5}\frac{\pi}{{12}}\right)}}}$ and $\displaystyle{\sin{{\left({5}\frac{\pi}{{12}}\right)}}}$ , such that:

$\displaystyle{z}={6}{\left({\cos{{\left({5}\frac{\pi}{{12}}\right)}}}+{i}\cdot{\sin{{\left({5}\frac{\pi}{{12}}\right)}}}\right)}$

$\displaystyle{\cos{{\left({5}\frac{\pi}{{12}}\right)}}}={\cos{{\left({6}\frac{\pi}{{12}}-\frac{\pi}{{12}}\right)}}}={\cos{{\left(\frac{\pi}{{2}}-\frac{\pi}{{12}}\right)}}}={\sin{{\left(\frac{\pi}{{12}}\right)}}}$

$\displaystyle{\sin{{\left(\frac{\pi}{{12}}\right)}}}={\sin{{\left(\frac{{\frac{\pi}{{6}}}}{{2}}\right)}}}=\sqrt{{\frac{{{1}-{\cos{{\left(\frac{\pi}{{6}}\right)}}}}}{{2}}}}$

$\displaystyle{\sin{{\left(\frac{\pi}{{12}}\right)}}}=\frac{{\sqrt{{{2}-\sqrt{{3}}}}}}{{2}}$

$\displaystyle{\sin{{\left({5}\frac{\pi}{{12}}\right)}}}={\cos{{\left(\frac{\pi}{{12}}\right)}}}=\frac{{\sqrt{{{2}+\sqrt{{3}}}}}}{{2}}$

$\displaystyle{z}={6}{\left(\frac{{\sqrt{{{2}-\sqrt{{3}}}}}}{{2}}+{i}\cdot\frac{{\sqrt{{{2}+\sqrt{{3}}}}}}{{2}}\right)}$

$\displaystyle{z}={3}{\left(\sqrt{{{2}-\sqrt{{3}}}}+{i}\cdot{\left(\sqrt{{{2}+\sqrt{{3}}}}\right)}\right)}$

Hence, the standard form of the given complex number is $\displaystyle{z}={3}{\left(\sqrt{{{2}-\sqrt{{3}}}}+{i}\cdot{\left(\sqrt{{{2}+\sqrt{{3}}}}\right)}\right)}.$

Blog

Wednesday, March 31, 2010

$\displaystyle{6}{\left({\cos{{\left(\frac{{{5}\pi}}{{12}}\right)}}}+{i}{\sin{{\left(\frac{{{5}\pi}}{{12}}\right)}}}\right)}$ Write the standard form of the complex number.

No comments:

Post a Comment

How does author Elie Wiesel use symbolism to contribute to the meaning of Night?