Please Help With This Trig Problem

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

In order to divide two complex numbers that are already in polar form, we can use a rule that makes this much faster than it looks. When you divide, you divide the two moduli and you subtract the angles. That is the whole problem, and it is worth memorizing because the alternative is multiplying by the conjugate and expanding everything out, which takes a page and gives the same thing.

So for your problem, the moduli are 5 and 2, and the angles are 220° and 115°. Dividing the moduli gives us:

5/2

and subtracting the angles, top minus bottom, gives us:

220° - 115° = 105°

Putting those back together:

5/2(cos 105° + i sin 105°)

which is choice A.

If you notice, the two wrong answers that are there to catch you are B and C. Choice B flipped the division and gave 2/5 instead of 5/2, which happens when you divide the bottom by the top by mistake, so always keep track of which one is the numerator. Choice C kept the 5/2 correctly but used 335°, and 335 is 220 plus 115, so that is the one you land on if you add the angles instead of subtracting them. Adding is what you do when you MULTIPLY two complex numbers in polar form, so those two rules sit right next to each other in your book and it is very easy to grab the wrong one. Choice D made both of those mistakes at once, multiplying the moduli to get 10 and adding the angles to get 335.

Remember, our purpose is to divide, so the pairing you want is divide the moduli and subtract the angles. A way to keep it straight is to think about what the modulus means. The modulus is the size of the number, so if you divide a number of size 5 by a number of size 2, the answer should be smaller than 5, and 5/2 is 2.5 while choice D's 10 is bigger. Just checking whether the size moved in a sensible direction rules out half the choices on these.

I also checked this the long way to be sure, by converting both numbers into a + bi form, dividing them out, and converting back, and it came out to a modulus of exactly 5/2 and an angle of exactly 105°, so A is right.

One more thing that is worth knowing for the rest of this unit. This same rule is where De Moivre's theorem comes from. If dividing subtracts angles and multiplying adds them, then raising to a power n multiplies the angle by n and raises the modulus to the n, so a problem asking you for something like the fifth power of a complex number is the same idea you just used here.

So, our final answer is A, 5/2(cos 105° + i sin 105°).

Answer 2

apply the rule; simply divide the constants and subtract the angles

5(cos220°+isin220°)/2(cos115°+isin115°)

(5/2)[(cos220-115)+isin(220-115)]

(5/2)(cos105°+isin105°) which is A.

Answer 3

Remember Euler's formula, e^(ix) = cos(x) + i*sin(x). So then we have:

5(cos(220) + i*sin(220))/(2(cos(115)+i*sin(115))) = 5e^(i*220)/2e^(i*115) = 5/2*e^(i*105) = 5/2(cos(105) + i*sin(105)).

Note: technically when using Euler's formula you're supposed to use radians instead of degrees, but it doesn't change the answer in this case because you're transforming it back.