Find the product 2(cos10degree+isin 10degree).4(cos20degree+isin20degree).Then express the result in rectangular form.

Answer 1

From series expansions, we can deduce that

cos(x) + i sin(x)
=1 - x²/2! + x⁴/4! - ...
+ x - x³/3! + x⁵/5!
=1 + (ix)²/2! + (ix)³/3! + (ix)x⁴/4! + (ix)⁵/5! + ...
= e^ix

Therefore
2(cos10degree+isin 10degree).4(cos20degree+isin20degree)
=2e^(iπ/18).4e^(iπ/9)
=2.4e^{(iπ/18)+(iπ/9)}
=8e^(iπ/6)
=8(e^iπ)^(1/6)
=8(-1)^(1/6)

Note:
e^iπ
= cos(π) + i sin(π)
= -1 + 0
= -1