Find the quotient of the complex numbers

Question

Find the quotient of the complex numbers: 

 

2*(sqrt 3)(cos(116 deg)+isin(116 deg))/2(cos(82 deg)+isin(82 deg))

Answer 1

Use De Moivre's formula, where the angle of numerator is reduced by the angle in the denominator and the "r" values in front of the (cos + i sin) part are divided.

Answer 2

I think that what makes this one a bit tricky is that is looks complex.  But if we use Euler's Equation, we can make this into a much simpler expression.  Have you studied Euler yet?  (It's pronounced "oiler"... maybe you remember your teacher mentioning this?)

 

according to Euler, cos(166^ο) + isin(166^ο) = e^166deg*ⁱ

 

sorry - couldn't figure out how to put a degree sign in an exponent :)

 

 

similarly the bottom part can convert to e^82deg*ⁱ

 

 

so the whole thing converts to :

 

2√3 e^166deg*i

2 e^82deg*i

 

The 2's cancel.  Then we can combine the exponents of 'e'.

 

166-82 = 84

 

√3 e^84deg*ⁱ

 

your teacher MAY want that back in polar form.  Do you know how to do Euler's formula in reverse, now that we've done it forwards?