WEBVTT

1
00:00:00.000 --> 00:00:05.968
Here is the problem: 2cos(x) squared plus 3cos(x) plus 1 equals 0.

2
00:00:06.418 --> 00:00:14.104
A periodic equation in x: it has infinitely many solutions, one family for each solution in a single period.

3
00:00:14.904 --> 00:00:16.378
Start with the equation.

4
00:00:16.828 --> 00:00:18.988
Rewrite as f(x) equals 0.

5
00:00:19.438 --> 00:00:33.463
The left side repeats every 2pi, so find the solutions in one period. f(x plus 2pi) equals f(x) for every x, so every solution is one of the solutions in [0, 2pi) plus a whole number of periods.

6
00:00:33.913 --> 00:00:37.013
Solutions in one period [0, 2pi).

7
00:00:37.813 --> 00:00:39.845
Add any whole number of periods.

8
00:00:40.645 --> 00:00:52.417
So the answer is x equals 2pi/3 plus 2pik, x equals pi plus 2pik, x equals 4pi/3 plus 2pik (k any integer).

9
00:00:52.868 --> 00:01:03.282
Each solution in one period was found numerically, matched to an exact value and confirmed by substitution; adding whole periods gives all solutions.

10
00:01:04.082 --> 00:01:13.532
x equals 2.0943951: left side equals 0, right side equals 0, which checks out.

11
00:01:13.982 --> 00:01:23.165
x equals 3.1415927: left side equals 0, right side equals 0, which checks out.

12
00:01:23.615 --> 00:01:32.659
x equals 4.1887902: left side equals 0, right side equals 0, which checks out.