WEBVTT

1
00:00:00.000 --> 00:00:03.622
Here is the problem: sin x equals 0.5.

2
00:00:04.072 --> 00:00:11.595
A periodic equation in x: it has infinitely many solutions, one family for each solution in a single period.

3
00:00:12.395 --> 00:00:13.904
Start with the equation.

4
00:00:14.354 --> 00:00:16.432
Rewrite as f(x) equals 0.

5
00:00:16.882 --> 00:00:31.023
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
00:00:31.473 --> 00:00:34.561
Solutions in one period [0, 2pi).

7
00:00:35.361 --> 00:00:37.369
Add any whole number of periods.

8
00:00:38.170 --> 00:00:47.690
So the answer is x equals pi/6 plus 2pik, x equals 5pi/6 plus 2pik (k any integer).

9
00:00:48.140 --> 00:00:58.299
Each solution in one period was found numerically, matched to an exact value and confirmed by substitution; adding whole periods gives all solutions.

10
00:00:59.099 --> 00:01:10.036
x equals 0.52359878: left side equals 0.5, right side equals 0.5, which checks out.

11
00:01:10.486 --> 00:01:21.179
x equals 2.6179939: left side equals 0.5, right side equals 0.5, which checks out.