Question

Divide. 9(cos(11π/6) +sin(11π/6))3√3(cos(π/4) +i sin(π/4)) Enter your answer by filling in the boxes. Enter all values as exact values in simplest form. (Cos () +i sin ()

Answer 1

SOLUTION

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the given expression

$\frac{9(cos(\frac{11x}{6})+isin(\frac{11x}{6}))}{3\sqrt{3}(cos(\frac{\pi}{4})+isin(\frac{\pi}{4}))}$

STEP 2: Simplify the expression

$3\sqrt{3}\left(\cos\left(\frac{\pi}{4}\right)+i\sin\left(\frac{\pi}{4}\right)\right)=3\sqrt{3}\left(i\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)$

STEP 3: Rewrite the expression

$=\frac{9\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{3\sqrt{3}\left(i\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)}$

Divide the numbers:

$\begin{gathered} \mathrm{Divide\:the\:numbers:}\:\frac{9}{3}=3 \\ =\frac{3\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{\sqrt{3}\left(i\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)} \end{gathered}$

STEP 4: Apply Radical rule

$\begin{gathered} \mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{a}=a^{\frac{1}{n}} \\ \sqrt{3}=3^{\frac{1}{2}} \\ =\frac{3\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{3^{\frac{1}{2}}\left(i\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)} \end{gathered}$

STEP 5: Apply Exponent rule

$\begin{gathered} \mathrm{Apply\:exponent\:rule}:\quad \frac{x^a}{x^b}=x^{a-b} \\ \frac{3^1}{3^{\frac{1}{2}}}=3^{1-\frac{1}{2}} \\ =\frac{3^{-\frac{1}{2}+1}\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}} \\ \mathrm{Subtract\:the\:numbers:}\:1-\frac{1}{2}=\frac{1}{2} \\ =\frac{3^{\frac{1}{2}}\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}} \end{gathered}$

STEP 6: Apply Radical rule

$\begin{gathered} \mathrm{Apply\:radical\:rule}:\quad \:a^{\frac{1}{n}}=\sqrt[n]{a} \\ 3^{\frac{1}{2}}=\sqrt{3} \\ =\frac{\sqrt{3}\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}} \\ \text{By Multiplication,} \\ i\frac{\sqrt{2}}{2}=\frac{\sqrt{2}i}{2} \\ =\frac{\sqrt{3}\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{\frac{\sqrt{2}}{2}+\frac{\sqrt{2}i}{2}} \end{gathered}$

STEP 7: Combine the fractions

$\begin{gathered} \frac{\sqrt{2}}{2}+\frac{\sqrt{2}i}{2}=\frac{\sqrt{2}+\sqrt{2}i}{2} \\ =\frac{\sqrt{3}\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{\frac{\sqrt{2}+\sqrt{2}i}{2}} \\ \mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{\frac{b}{c}}=\frac{a\cdot \:c}{b} \\ \frac{\sqrt{3}\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)\cdot \:2}{\sqrt{2}+\sqrt{2}i} \end{gathered}$

STEP 8: Factor out common term

$\begin{gathered} =\frac{\sqrt{3}\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)\cdot \:2}{\sqrt{2}\left(1+i\right)} \\ =\frac{\sqrt{3}\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)\sqrt{2}}{1+i} \end{gathered}$

By simplification,

$=\frac{\sqrt{6}\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{1+i}$

STEP 9: Rationalize

$\frac{\sqrt{6}\left(\cos\left(\frac{11x}{6}\right)+i\sin\left(\frac{11x}{6}\right)\right)}{1+i}=\frac{\sqrt{6}\left(1-i\right)\left(\cos\left(\frac{11x}{6}\right)+i\sin\left(\frac{11x}{6}\right)\right)}{2}$

STEP 10: Write the answer in the required form

$\begin{gathered} \frac{\sqrt{6}\left(1-i\right)\left(\cos \left(\frac{11x}{6}\right)+i\sin \left(\frac{11x}{6}\right)\right)}{2} \\ \frac{\sqrt{6}\left(1-i\right)}{2}\times\left(\cos\left(\frac{11x}{6}\right)+i\sin\left(\frac{11x}{6}\right)\right) \\ =\sqrt{\frac{3}{2}}-\sqrt{\frac{3}{2}}i\times(\cos(\frac{11x}{6})+\imaginaryI\sin(\frac{11x}{6})) \end{gathered}$

ANSWER:

$\sqrt{\frac{3}{2}}-\sqrt{\frac{3}{2}}i\cdot\left(\cos\left(\frac{11x}{6}\right)+i\sin\left(\frac{11x}{6}\right)\right)$