WEBVTT
00:00:01.030 --> 00:00:14.400
Find the first five terms of the sequence whose πth term is given by π sub π equals cos 11π over six π, where π is greater than or equal to one.
00:00:15.380 --> 00:00:28.740
In order to calculate the first five terms of any sequence starting with π equals one, we substitute the values one, two, three, four, and five into our expression.
00:00:29.850 --> 00:00:36.810
π sub one is therefore equal to cos of 11 multiplied by one over six π.
00:00:38.020 --> 00:00:44.460
π sub two is equal to cos of 11 multiplied by two divided by six π.
00:00:45.510 --> 00:00:53.280
These simplify to cos of 11 over six π and cos of 22 over six π, respectively.
00:00:54.570 --> 00:01:02.160
π three is equal to cos of 33 over six π as 11 multiplied by three is 33.
00:01:03.040 --> 00:01:13.170
Likewise, π sub four is equal to cos of 44 over six π, and π sub five is equal to cos 55 over six π.
00:01:14.360 --> 00:01:17.750
At this stage, we could proceed in a few different ways.
00:01:18.190 --> 00:01:25.170
We could just type all five expressions into the calculator, ensuring that we were in radian mode.
00:01:26.440 --> 00:01:30.070
This would give us the first five terms of the sequence.
00:01:30.980 --> 00:01:41.840
Alternatively, we could cancel the fractions where possible; 22 over six, 33 over six, and 44 over six can all be canceled.
00:01:42.830 --> 00:01:50.190
Instead of doing either of these things, we will look at the properties of the cosine graph and some of our special angles.
00:01:51.120 --> 00:01:55.870
This will allow us to calculate the answers without using a calculator.
00:01:56.980 --> 00:02:02.380
We recall that cos of 30 degrees is equal to root three over two.
00:02:03.330 --> 00:02:12.830
As 180 degrees is equal to π radians, then cos of π by six radians is also equal to root three over two.
00:02:13.870 --> 00:02:18.840
cos of π over three or 60 degrees is equal to one-half.
00:02:19.590 --> 00:02:27.220
As π over three is equivalent to two π over six, cos of two π over six equals one-half.
00:02:28.320 --> 00:02:34.300
We also know that cos of π over two or 90 degrees is equal to zero.
00:02:35.580 --> 00:02:39.950
This means that cos of three π over six equals zero.
00:02:41.120 --> 00:02:50.370
cos of four π over six is equal to negative one-half, and cos of five π by six is negative root three over two.
00:02:51.090 --> 00:03:06.940
As the cosine graph is periodic with a period of two π, we can calculate the value of cos 11 over six π, cos 22 over six π, and so on using these values and the pattern.
00:03:07.750 --> 00:03:17.010
By considering our CAST diagram, we can find the cosine of all multiples of π over six that are equal to root three over two.
00:03:17.960 --> 00:03:23.140
Two π minus π over six is equal to 11π over six.
00:03:24.170 --> 00:03:36.810
As already mentioned, as the cosine graph is periodic and has a period of two π, we can add two π to both of these values to find an infinite number of solutions.
00:03:37.790 --> 00:03:46.120
cos of 13π over six and cos of 23π over six are both equal to root three over two.
00:03:47.190 --> 00:03:56.480
As the first value π sub one in our sequence was cos 11 over six π, this is equal to root three over two.
00:03:57.740 --> 00:04:06.760
Repeating this process, we see that cos of two over six π and cos of 10 over six π are equal to one-half.
00:04:07.690 --> 00:04:18.580
Once again, we can add two π or 12 over six π to both of these, giving us 14 over six π and 22 over six π.
00:04:19.680 --> 00:04:25.830
π sub two, cos of 22 over six π, is equal to one-half.
00:04:26.700 --> 00:04:45.520
We could continue this method to calculate cos of 33 over six π which equals zero, cos of 44 over six π which is negative one-half, and cos of 55 over six π which is equal to negative root three over two.
00:04:46.620 --> 00:05:05.470
As previously mentioned, there is no problem with typing the five expressions into our calculator in radian mode to get the first five terms root three over two, one-half, zero, negative one-half, and negative root three over two.
00:05:06.520 --> 00:05:14.060
It is important, however, to recall our special angles as these will save us time when dealing with some problems.