WEBVTT

1
00:00:00.000 --> 00:00:03.100
Here is the problem: sin(2x) equals 0.

2
00:00:03.550 --> 00:00:11.062
A periodic equation in x: it has infinitely many solutions, one family for each solution in a single period.

3
00:00:11.862 --> 00:00:13.371
Start with the equation.

4
00:00:13.821 --> 00:00:15.934
Rewrite as f(x) equals 0.

5
00:00:16.384 --> 00:00:29.956
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
00:00:30.406 --> 00:00:33.169
Solutions in one period [0, pi).

7
00:00:33.969 --> 00:00:35.954
Add any whole number of periods.

8
00:00:36.754 --> 00:00:42.989
So the answer is x equals pik, x equals pi/2 plus pik (k any integer).

9
00:00:43.439 --> 00:00:53.667
Each solution in one period was found numerically, matched to an exact value and confirmed by substitution; adding whole periods gives all solutions.

10
00:00:54.467 --> 00:01:01.236
x equals 0: left side equals 0, right side equals 0, which checks out.

11
00:01:01.686 --> 00:01:10.986
x equals 1.5707963: left side equals 0, right side equals 0, which checks out.