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

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

STEP 1: Write the given expression

`(9(cos((11x)/(6))+isin((11x)/(6))))/(3√(3)(cos((\pi)/(4))+isin((\pi)/(4))))`

STEP 2: Simplify the expression

`3√(3)\left(\cos\left((\pi)/(4)\right)+i\sin\left((\pi)/(4)\right)\right)=3√(3)\left(i(√(2))/(2)+(√(2))/(2)\right)`

STEP 3: Rewrite the expression

`=(9\left(\cos \left((11x)/(6)\right)+i\sin \left((11x)/(6)\right)\right))/(3√(3)\left(i(√(2))/(2)+(√(2))/(2)\right))`

Divide the numbers:

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

STEP 4: Apply Radical rule

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

STEP 5: Apply Exponent rule

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

STEP 6: Apply Radical rule

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

STEP 7: Combine the fractions

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

STEP 8: Factor out common term

`\begin{gathered} =(√(3)\left(\cos \left((11x)/(6)\right)+i\sin \left((11x)/(6)\right)\right)\cdot \:2)/(√(2)\left(1+i\right)) \\ =(√(3)\left(\cos \left((11x)/(6)\right)+i\sin \left((11x)/(6)\right)\right)√(2))/(1+i) \end{gathered}`

By simplification,

`=(√(6)\left(\cos \left((11x)/(6)\right)+i\sin \left((11x)/(6)\right)\right))/(1+i)`

STEP 9: Rationalize

`(√(6)\left(\cos\left((11x)/(6)\right)+i\sin\left((11x)/(6)\right)\right))/(1+i)=(√(6)\left(1-i\right)\left(\cos\left((11x)/(6)\right)+i\sin\left((11x)/(6)\right)\right))/(2)`

STEP 10: Write the answer in the required form

`\begin{gathered} (√(6)\left(1-i\right)\left(\cos \left((11x)/(6)\right)+i\sin \left((11x)/(6)\right)\right))/(2) \\ (√(6)\left(1-i\right))/(2)*\left(\cos\left((11x)/(6)\right)+i\sin\left((11x)/(6)\right)\right) \\ =\sqrt{(3)/(2)}-\sqrt{(3)/(2)}i*(\cos((11x)/(6))+\imaginaryI\sin((11x)/(6))) \end{gathered}`

ANSWER:

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