Converting from Exponential to Cartesian form for complex numbers?

Question

I have a set of q's on this... once I know the method for one I'll be able to do them all but currently I don't get it. Help greatly appreciated.

Write (1+23i)e^23*pi*i in the form x + iy with x, y real numbers.

Answer 1

Remember that e^(iw) =cos(w) + isin(w), so that the exponential form for complex numbers is just another way of writing them in polar coordinates (r,w), where r is the modulus and w the argument.

In the example, w=23*pi and cos(23pi) = -1 whilst sin(23pi) = 0 so
e^(23pi*i) = -1
and (1+23i)*(-1) = -1 -23i.

Answer 2

Complex Numbers Exponential Form

Answer 3

(1+23i)e^23*pi*i, strictly, means (1+23i)*(e^23)*pi*i which is:
-23*pi*e^23 + (pi*e^23)i.
But I would guess that you mean: (1+23i)e^(23*pi*i).
Euler's formula states that, for any real number x,
e^(ix) = cos(x) + i*sin(x)
Please memorize this Euler's formula. It is very useful.
Hence: (1+23i)e^(23*pi*i) = (1+23i)*(-1+0) = -1 - 23i.

Answer 4

in case you plotted it, you've already were given 1/2 of the wrestle executed! First, you should determine out the size, r, of your volume. seem at your plotted factor. are you able to tell how a thoughts it really is from the starting up? (hint: there will be a objective your instructor chosen 3 and four.) also, for the list, r is continually presumed to be constructive once you're doing polar coordinates - undesirable distances make no experience. So, now that you've r, you should locate theta. you've already got a technique of a threat values of theta: Quadrant a million: 0 to ninety degrees Quadrant 2: ninety to at least one hundred 80 degrees Quadrant 3: one hundred 80 to 270 degrees Quadrant 4: 270 to 360 degrees Theta actually is authorized to flow over 360 degrees - 360 to 450 degrees ought to start up again in quadrant a million. you'll locate theta in really some fantastic thoughts. you should have already were given a distance of three and four marked on your diagram. Draw an attitude, theta, that is going between the x-axis and your aspect. Then, write this down: tan theta = ??? / ??? Use your SOH-CAH-TOA to seek out the most options-blowing ratio. you should use sine or cosine too, notwithstanding you should ought to understand the hypotenuse. (obviously, you've already got been given the hypotenuse in case you probably did it so as.) therefore, stumble on that theta gives you what number degrees below the x-axis your factor is. You obviously ought to favor to flow round any opposite direction - starting up from the +x axis, you should flow from quadrant a million to 2 to three to the arrow you drew. it really is definitely 360 degrees MINUS your attitude.

Answer 5

Hi,

e^(πi) = -1 so substituting this into your problem simplifies it to:

(1 + 23i)e^(23πi) = (1 + 23i)[e^(πi)]^(23) = (1 + 23i)[-1]^(23) =

(1 + 23i)[-1] = -1 - 23i

I hope that helps!! :-)