WEBVTT
00:00:01.810 --> 00:00:16.030
Given that π one is equal to two multiplied by cos 90 minus π sin 90 and π two equals four multiplied by sin 30 plus π cos 30, find π one π two, giving your answer in exponential form.
00:00:17.450 --> 00:00:24.570
A complex number is in polar form if it looks like this: π is equal to π cos π plus π sin π.
00:00:25.400 --> 00:00:29.190
Notice that our complex number for π two looks slightly different.
00:00:29.330 --> 00:00:33.760
Weβre going to need to perform some clever manipulation to make our equations look the same.
00:00:34.890 --> 00:00:46.340
Remember the sine and cosine graphs are horizontal translations of each other such that sine of π is equal to cos of 90 minus π and cos of π is equal to sine of 90 minus π.
00:00:47.760 --> 00:00:54.530
That means that sine of 30 is equal to cos of 90 minus 30, which is equal to cos of 60.
00:00:55.260 --> 00:01:02.120
Similarly, cos of 30 is equal to sin of 90 minus 30, which is equal to sin of 60.
00:01:02.910 --> 00:01:07.140
And π two can be written as four cos 60 plus π sin 60.
00:01:07.850 --> 00:01:10.820
Normally, we might try to use the product formula here.
00:01:11.180 --> 00:01:23.100
But since our first complex number has a negative coefficient for π sin π and our second has a positive coefficient for π sin π, instead weβll first convert them into exponential form.
00:01:24.060 --> 00:01:30.060
Recall Eulerβs identity says π to the plus or minus ππ is equal to cos π plus π sin π.
00:01:30.340 --> 00:01:41.250
And we can extend that and say that ππ to the plus or minus ππ is equal to π multiplied by cos π plus π sin π, where π is the modulus and π is the argument.
00:01:41.550 --> 00:01:44.990
But remember this only holds true if π is in radians.
00:01:45.590 --> 00:01:49.580
Letβs use this information to write our complex numbers in exponential form.
00:01:50.370 --> 00:01:54.160
The modulus of π one is two and the argument is 90.
00:01:54.760 --> 00:01:59.590
We can convert from degrees to radians by multiplying by π over 180.
00:02:00.420 --> 00:02:04.570
And doing so and we can see that the argument for π one is π over two.
00:02:05.520 --> 00:02:14.880
Since the coefficient for π sin π is negative, π one can be written in exponential form as two π to the negative π over two π.
00:02:15.780 --> 00:02:20.140
The modulus of π two is four and π is 60 degrees.
00:02:20.790 --> 00:02:25.460
Once again, weβll multiply this by π over 180 to turn into radians.
00:02:26.080 --> 00:02:28.530
And doing so, we get π over three.
00:02:29.160 --> 00:02:33.650
And we can rewrite π two as four π to the π over three π.
00:02:34.410 --> 00:02:38.480
The question is asking us to find the product of these two numbers, to multiply them together.
00:02:38.770 --> 00:02:40.170
Weβll do that in the usual way.
00:02:40.910 --> 00:02:46.780
Itβs two π to the negative π over two π multiplied by four π to the π over three π.
00:02:47.520 --> 00:02:49.770
Two multiplied by four is eight.
00:02:50.460 --> 00:02:53.930
And when we multiply two numbers with the same base, we add the exponents.
00:02:53.930 --> 00:02:58.110
So thatβs π to the negative π over two plus π over three π.
00:02:58.760 --> 00:03:05.080
And we can see that the exponential form of π one π two is eight π to the negative π over six π.
00:03:05.670 --> 00:03:12.680
Notice though that each of the exponents in the possible answers to this question have a positive coefficient of π.
00:03:13.330 --> 00:03:22.460
Since the imaginary exponential is periodic with a period of two π, we can add two π to negative π over six until we find a positive value.
00:03:23.230 --> 00:03:27.450
Negative π over six plus two π is 11π over six.
00:03:28.110 --> 00:03:32.700
And π one π two is, therefore, eight π to the 11π over six π.