Complex, yet Simple
If cos ( 2 0 1 4 θ ) + i sin ( 2 0 1 4 θ ) = x 2 0 1 4 , find x e i θ .
Clarification : i = − 1 .
The answer is 1.
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2 solutions
Nicely done, David Lee, you made it look so complicated, yet it was so simple. First solver!
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Well, 2 theorems conquers everything. :P
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What about 3?
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@Sharky Kesa – Which 3? De Moivre and Euler pretty much condones everything. Unless you are talking about the reducing fractions rule... :P
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@David Lee – Reducing Fractions and an extended variation of Pythagoras' Theorem.
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@Sharky Kesa – Well, the extended variation is De Moivre.
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@David Lee – Isn't the extended version the Law of Cosines?Not De Moivre?
Very Troll. Much Wow. So Mental Math. Very One.
Technically this question will give answer in the form of c o s ( 1 0 0 7 n π ) + i s i n ( 1 0 0 7 n π )
for n = 0 , 1 , 2 , . . . , 1 0 0 6 .
By De Moivre's Theorem, x = cos ( θ ) + i sin ( θ ) . By Euler's Identity,We see that e i θ = x . Therefore, x x = 1 .