Find the Quotient

Question

5(cos 220° + i sin 220°)/2(cos 115° + i sin 115°)

A. 5/2(cos 105° + i sin 105°)

B. 2/5(cos 105° + i sin 105°)

C. 5/2(cos 335° + i sin 335°)

D. 10(cos 335° + i sin 335°)

Answer 1

I did this earlier

divide the coefficients 5 and 2 and subtract the degrees

(5/2)(cos105°+isin105°) which is A.

Answer 2

Solving this problem involves a combination of exponent rules and a really crazy thing called Euler's formula.

Euler's formula gets great reviews from famous mathematicians and scientists over the years and I hear it is really important in engineering and science. According to Wikipedia, Richard Feynman referred to it as "the most remarkable formula in mathematics."

I can't actually tell you much about why it is so special. For your problem, all you need to know is what Euler's formula is...

cos(x) + i(sin(x)) = e^ix

Keep in mind that the i you are dealing with is the imaginary number i.

Steps for your problem:

1. Take a look at how you can make both the top and the bottom of your division problem a lot simpler with Euler's formula. For example the top of the fraction can change to cos(220) + i(sin(220)) = e²²⁰ⁱ
2. Factor out 5 from the numerator and 2 from the denominator. This can be set aside as a coefficient of 5/2.
3. Use good old exponent rules to simplify your expression. Remember that if you have the same base, like e, and you are dividing, you can subtract the powers. You're on the right track if you end up doing e^{220i - 115i}.
4. Notice that the answer choices don't involve anything about the number e. Use Euler's formula again to find which expression of cos(x) + i sin(x) is equivalent to your work so far.

Thanks for your question because I definitely learned something as I did the research to answer it! Good luck.