WEBVTT

1
00:00:00.000 --> 00:00:03.378
Here is the problem: cos(2x) equals 0.

2
00:00:03.829 --> 00:00:11.376
A periodic equation in x: it has infinitely many solutions, one family for each solution in a single period.

3
00:00:12.176 --> 00:00:13.674
Start with the equation.

4
00:00:14.124 --> 00:00:16.225
Rewrite as f(x) equals 0.

5
00:00:16.675 --> 00:00:30.108
The left side repeats every pi, so find the solutions in one period. f(x plus pi) equals f(x) for every x, so every solution is one of the solutions in [0, pi) plus a whole number of periods.

6
00:00:30.558 --> 00:00:33.286
Solutions in one period [0, pi).

7
00:00:34.086 --> 00:00:36.094
Add any whole number of periods.

8
00:00:36.895 --> 00:00:45.161
So the answer is x equals pi/4 plus pik, x equals 3pi/4 plus pik (k any integer).

9
00:00:45.611 --> 00:00:55.956
Each solution in one period was found numerically, matched to an exact value and confirmed by substitution; adding whole periods gives all solutions.

10
00:00:56.755 --> 00:01:06.217
x equals 0.78539816: left side equals 0, right side equals 0, which checks out.

11
00:01:06.667 --> 00:01:16.141
x equals 2.3561945: left side equals 0, right side equals 0, which checks out.