Sine: Equation

How to solve the sine equation sin x = k: general solution formula, 1 example and its solution.

Formula

Formula

For sin x = k,
if one of the solution is θ,
then the general solution is
x = nπ + (-1)n⋅θ.
(n is an integer.)

Example

Example

Solution

See sin x = 1/2.

Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (2).

So draw a right triangle on a coordinate plane
whose opposite side is 1
and whose hypotenuse is 2.

Find the missing side
by using the Pythagorean theorem:
[base]2 + 12 = 22.

Then the base is √3.

This is a right triangle
whose sides are 1, √3, and 2.

So this is a 30-60-90 triangle.

So the central angle is, 30º, π/6.

Radian Measure

π/6 is the angle
that starts from 3 o'clock position.

So θ = π/6.

Then the general solution of sin x = 1/2 is
x = nπ + (-1)n⋅[π/6].

Find the x values
that are in (0 ≤ x ≤ 2π).

n = 0

x = 0⋅π + (-1)0⋅[π/6]
= π/6

This is in (0 ≤ x ≤ 2π).

n = 1

x = 1⋅π + (-1)1⋅[π/6]
= 5π/6

This is also in (0 ≤ x ≤ 2π).

x = π/6 and x = 5π/6
are in (0 ≤ x ≤ 2π).

So write x = π/6, 5π/6.

So x = π/6, 5π/6 is the answer.