Respuesta :
The given complex number is:
[tex](2cis( \frac{ \pi }{9}))^{3} \\ \\ [/tex]
cis(x) means cos(x) + i sin(x). So the above expression can be written as:
[tex] (2(cos \frac{ \pi }{9}+isin \frac{ \pi }{9}))^{3} [/tex]
Using the DeMoivres Theorem we can simplify the expression as:
[tex](2(cos \frac{ \pi }{9}+isin \frac{ \pi }{9}))^{3} \\ \\ =2^{3}(cos(3* \frac{ \pi }{9})+isin(3* \frac{ \pi }{9})) \\ \\ =8(cos \frac{ \pi }{3}+isin \frac{ \pi }{3}) \\ \\ =8( \frac{1}{2}+i \frac{ \sqrt{3} }{2}) \\ \\ =4+i4 \sqrt{3} \\ \\ =4+4 \sqrt{3}i [/tex]
So, option A gives the correct answer
[tex](2cis( \frac{ \pi }{9}))^{3} \\ \\ [/tex]
cis(x) means cos(x) + i sin(x). So the above expression can be written as:
[tex] (2(cos \frac{ \pi }{9}+isin \frac{ \pi }{9}))^{3} [/tex]
Using the DeMoivres Theorem we can simplify the expression as:
[tex](2(cos \frac{ \pi }{9}+isin \frac{ \pi }{9}))^{3} \\ \\ =2^{3}(cos(3* \frac{ \pi }{9})+isin(3* \frac{ \pi }{9})) \\ \\ =8(cos \frac{ \pi }{3}+isin \frac{ \pi }{3}) \\ \\ =8( \frac{1}{2}+i \frac{ \sqrt{3} }{2}) \\ \\ =4+i4 \sqrt{3} \\ \\ =4+4 \sqrt{3}i [/tex]
So, option A gives the correct answer
