WEBVTT
00:00:00.880 --> 00:00:06.230
Put π§ equals five root three π to the π over three π in algebraic form.
00:00:06.980 --> 00:00:10.070
Weβve been given this complex number π§ in exponential form.
00:00:10.070 --> 00:00:20.710
Thatβs the form π§ equals ππ to the ππ, where π is the absolute value of the complex number and π is its argument measured in radians.
00:00:21.570 --> 00:00:31.650
And we know another way that we can express a complex number is in algebraic form, that is, π§ equals π plus ππ, which is what weβre going to convert this to.
00:00:32.380 --> 00:00:37.040
But in order to do this, weβll start by expressing π§ in its polar form.
00:00:37.650 --> 00:00:42.570
Thatβs the form π§ equals π multiplied by cos π plus π sin of π.
00:00:43.160 --> 00:00:44.500
But why are we doing this?
00:00:44.950 --> 00:00:54.000
Well, by writing it in this way, we can then distribute the parentheses to give us π§ equals π cos of π add ππ sin of π.
00:00:54.450 --> 00:01:05.090
And from here, π cos of π is the real part of the complex number, π, and π sin of π is the imaginary part of the complex number, π.
00:01:05.780 --> 00:01:10.210
And that gives us the algebraic form of π§ equals π plus ππ.
00:01:10.860 --> 00:01:14.440
So letβs begin by writing this complex number in its polar form.
00:01:14.990 --> 00:01:24.790
We can see just by inspection that the modulus of the complex number π equals five root three and that the argument π is equal to π over three.
00:01:25.300 --> 00:01:39.480
So, using our values of π and π and the general polar form for a complex number, we have that our complex number can be written as π§ equals five root three multiplied by cos of π over three plus π sin of π over three.
00:01:40.130 --> 00:01:45.080
But letβs actually evaluate the values of cos of π over three and sin of π over three.
00:01:45.810 --> 00:01:55.720
Based on the unit circle, we can find that cos of π over three equals one over two, and we also get that sin of π over three equals root three over two.
00:01:56.450 --> 00:02:03.230
So, this gives us π§ equals five root three multiplied by one over two add root three over two π.
00:02:04.090 --> 00:02:13.020
Distributing the parentheses then gives us π§ equals five root three over two add five root three multiplied by root three over two π.
00:02:13.880 --> 00:02:28.950
But we know that root three multiplied by root three just gives us three and that five multiplied by three gives us 15 so that then gives us a final answer of π§ equals five root three over two add 15 over two π.
00:02:29.880 --> 00:02:42.580
So, by taking a complex number in exponential form and working out the value of π and π then converting the complex number into polar form, we were then able to convert it into algebraic form.