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Polar form: $\vert z \vert \big(\cos\theta + i\sin\theta \big)$

$$\begin{aligned}z^2 &= \vert 12^2 + 5^2 \vert\\ z &= \vert 13 \vert \\ \arctan \frac{5}{12} &= 22.61^\circ\\ z &= |13| \big(\cos(22.61^\circ) + i\sin(22.61^\circ)\big)\end{aligned}$$

Up until now my textbook has only shown answers with $\theta$ being in radian form. Is it acceptable to write the polar form with $\theta$ in degrees like what I did above?

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  • $\begingroup$ It should be noted that $arctan(5/12)$ is approximately $22.6^o$ $\endgroup$ – J. W. Tanner Feb 6 at 21:18
  • $\begingroup$ @EvanKim Be careful, this $z^2 = \vert 12^2 + 5^2 \vert$ is wrong. Similarly $z=|13|.$ $\endgroup$ – user376343 Feb 7 at 9:02
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Of course it's fine; degrees and radians are just different ways of representing the same quantity. It's just that radians are more usually used due to various reasons. See this expository article on the subject. Since $|13|=13$, you can also just write the result as $$13\cos 22.61^\circ+(13\sin 22.61^\circ)i.$$

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  • $\begingroup$ @EvanKim In my opinion, if the form requested is $|z|(\cos\theta + i\sin\theta)$ then the intended answer is $|13|(12/13 + i5/13).$ $\endgroup$ – user2661923 Feb 6 at 22:28
  • $\begingroup$ @user2661923 usually, such questions ask to put a complex number given in polar in Cartesian form, which is $x+iy$. It's quite weird to have the magnitude put outside. It's even weirder to write $|13|$ when it can be immediately simplified to $13$. If the OP is requiring this particular form, then they will have to specify. $\endgroup$ – YiFan Feb 6 at 22:30
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    $\begingroup$ I was guided by the 1st line of the OP's query. Absent that, my inclination would have been: $\;\theta\; =$ any angle whose cos is 12/13 and whose sin is 5/13, and the polar form is $\;13e^{i\theta}.$ $\endgroup$ – user2661923 Feb 6 at 22:34
  • $\begingroup$ How would you convert $22.61^\circ$ to radians? $\endgroup$ – Evan Kim Feb 6 at 22:44
  • $\begingroup$ @EvanKim $180^\circ=\pi$. Hence $1^\circ=\pi/180$, so... $\endgroup$ – YiFan Feb 6 at 22:58
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Writing the polar form using degrees instead of radians is legitimate, but not recommended. Radians are the better choice overall: one of the main reason being that they are plain numbers (https://en.wikipedia.org/wiki/Dimensionless_quantity) whereas degrees have a physical dimension.

That said, your notation is wrong. I corrected it:

$$|z|^2 = \vert 12^2 + 5^2 \vert$$ $$|z| = 13 $$ $$\theta=\arctan\left(\frac{5}{12}\right) = 22.61^\circ$$ $$z = 13 \big(\cos(22.61^\circ) + i\sin(22.61^\circ)\big)$$

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  • $\begingroup$ How would you convert $22.61^\circ$ to radians? $\endgroup$ – Evan Kim Feb 6 at 22:44
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    $\begingroup$ It is equivalent to $\frac{22.61}{180}\pi$ radians $\endgroup$ – b00n heT Feb 7 at 6:12
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    $\begingroup$ oh okay, I thought you had to do 22.61/180 and write that as the answer. Your answer doesn't look very messy at all $\endgroup$ – Evan Kim Feb 7 at 15:29

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