Question

How do you solve `Sin(2x)= 1/2`?

Answer 1

The smallest solutions are `x = 15` degrees and `x = 75` degrees. Other solutions can be formed by adding or subtracting multiples of `180`.

Picture an equilateral triangle with sides of length 1. Since the internal angles must add up to 180 degrees, they are each 60 degrees. Now cut it in two to produce two right-angled triangles. These will have internal angles of 30, 60 and 90 degrees.

The length of the shortest side is `1/2`. This side is opposite the smallest angle - 30 degrees. The length of the hypotenuse is 1, so `sin(30) = 1/2`. So if `x = 15`, `sin(2x) = sin(30) = 1/2`.

The next angle beyond `30` degrees such that `sin theta = 1/2` is `150 = 180 - 30`, yielding `x = 75`.