Hi, i am new to this website : ) i'm currently taking an online trig class on de moivre's theorem and i don't understand it at all! the question is: write each expression in the standard form for a complex number, a+bi.
1. [3cos(27))+isin(27)]^5
2. [2(cos(40))+isin(40)]^6
any info regarding this question would be extremely ! !

De Moivre's theorem uses this general formula z = r(cos α + i sin α) that is where we can have the form a + bi. If the given is raised to a certain number, then the r is raised to the same number while the angles are being multiplied by that number.

For 1) [3cos(27))+isin(27)]^5 we first apply the concept I mentioned above where it becomes

[3^5cos(27*5))+isin(27*5)] and then after simplifying we get, [243 (cos (135) + isin (135))] 

it is then further simplified to 243 (-1/ √2) + 243i (1/√2) = -243/√2 + 243/√2 i
and that is the answer.

For 2) [2(cos(40))+isin(40)]^6, we apply the same steps in 1)

[2^6(cos(40*6))+isin(40*6)],

[64(cos(240))+isin(240)] = 64 (-1/2) + 64i (-√3 /2)

And the answer is -32 -32 √3 i

Summary:
1) -243/√2 + 243/√2 i
2)-32 -32 √3 i

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