Find the trigonometric form of the complex number shown below.

Question

Quizplus · Accepted Answer

The number is in exponential form $re^{i	heta}$, where $r=11$ and $	heta=\frac{5\pi}{6}$. To see this, we convert the hexadecimal notation into binary: $1\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{e}6_{{\mathrm{base}}_{16}}=0001\ 1110\ 1010\ 1100\ 1100\ 1110\ 0110\ 0001_{{\mathrm{base}}_{2}}$, and then we split this into the sign, exponent, and mantissa as follows:

$	ext{sign}=0$
$	ext{exponent}=01111110_{{\mathrm{base}}_{2}}=126_{10}$
$	ext{mantissa}=1.\overline{10101100\ 11001110\ 01100001}_{{\mathrm{base}}_{2}}$

Therefore, the number in exponential form is $11e^{i	heta}$, where $	heta=\frac{\pi}{2}+\arctan(1.01\overline{0101100110011100110001}_{{\mathrm{base}}_{2}})\approx \frac{5\pi}{6}$. Finally, we can round the binary expansion of the argument to $1.01\overline{010}=\frac{5}{6}$, which explains why the correct answer is choice $\boxed{	extbf{(B)}}$.

Find the Trigonometric Form of the Complex Number Shown Below