How do you simplify #3(cos(pi/6)+isin(pi/6))div4(cos((2pi)/3)+isin((2pi)/3))# and express the result in rectangular form?

Question

Ariana Alvarez · Accepted Answer

The answer is #=-3/4i#

There are 2 ways for the simplification

#cos(pi/6)=sqrt3/2#

#sin(pi/6)=1/2#

#cos(2pi/3)=-1/2#

#sin(2pi/3)=sqrt3/2#

#i^2=-1#

#(3(cos(pi/6)+isin(pi/6)))/(4(cos(2pi/3)+isin(2pi/3)))#

#=3/4(sqrt3/2+i*1/2)/(-1/2+isqrt3/2)#

#=3/4((sqrt3/2+i*1/2)(-1/2-isqrt3/2))/((-1/2+isqrt3/2)(-1/2-isqrt3/2))#

#=3/4(-sqrt3/4-i3/4-i/4+sqrt3/4)/(1/4+3/4)#

#=3/4(-i)#

We can also use #costheta+isintheta=e^(itheta)#

#cos(pi/6)+isin(pi/6)=e^(ipi/6)#

#cos(2pi/3)+isin(2pi/3)=e^(2ipi/3)#

#:.(3(cos(pi/6)+isin(pi/6)))/(4(cos(2pi/3)+isin(2pi/3)))=3/4e^(ipi/6)/e^(2ipi/3)#

#=3/4(e^(ipi(1/6-2/3)))#

#=3/4e^(-ipi/2)#

#=3/4(cos(-pi/2)+isin(-pi/2))#

#=3/4*(0-i)#

#=-(3i)/4#

Serenity Johnson · Answer

#3(cos(pi/6)+isin(pi/6))-:4(cos(2pi/3)+isin(2pi/3))=-3/4i# Given two complex numbers #z_1=r_1(cosalpha+isinalpha)# and #z_2=r_2(cosbeta+isinbeta)# #z_1-:z_2=r_1/r_2(cos(alpha-beta)+iin(alpha-beta))# #3(cos(pi/6)+isin(pi/6))-:4(cos(2pi/3)+isin(2pi/3))# = #3/4(cos(pi/6-(2pi)/3)+isin(pi/6-(2pi)/3))# = #3/4(cos(pi/6-(4pi)/6)+isin(pi/6-(4pi)/6))# = #3/4(cos(-(3pi)/6)+isin(-(3pi)/6))# = #3/4(cos(-pi/2)+isin(-pi/2))# = #3/4(cos(pi/2)-isin(pi/2))# = #3/4(0-i)# = #-3/4i#

Xavier Alonso · Answer

undefined To simplify $\frac{3(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6}))}{4(\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}))}$ and express the result in rectangular form, follow these steps:

1. Use the properties of complex numbers to simplify the expression:
   $$\frac{3(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6}))}{4(\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}))}$$
   
2. Divide the real and imaginary parts separately:
   $$\frac{3\cos(\frac{\pi}{6})}{4\cos(\frac{2\pi}{3})} + \frac{3i\sin(\frac{\pi}{6})}{4i\sin(\frac{2\pi}{3})}$$

3. Simplify each part:
   $$\frac{3 \cdot \frac{\sqrt{3}}{2}}{4 \cdot (-\frac{1}{2})} + \frac{3i \cdot \frac{1}{2}}{4i \cdot (-\frac{\sqrt{3}}{2})}$$

4. Perform the arithmetic:
   $$\frac{-\frac{3\sqrt{3}}{4}}{2} + \frac{-\frac{3\sqrt{3}i}{8}}{-\frac{2\sqrt{3}i}{2}}$$
   
5. Simplify further:
   $$\frac{-3\sqrt{3}}{4} - \frac{-3\sqrt{3}i}{8} \cdot \frac{2}{-2\sqrt{3}i}$$
   $$= \frac{-3\sqrt{3}}{4} - \frac{-3\sqrt{3}i}{4}$$

6. Combine like terms:
   $$= \frac{-3\sqrt{3}}{4} + \frac{3\sqrt{3}i}{4}$$

7. Rewrite in rectangular form:
   $$= \frac{-3\sqrt{3} + 3\sqrt{3}i}{4}$$

Therefore, the simplified expression in rectangular form is $\frac{-3\sqrt{3} + 3\sqrt{3}i}{4}$.