# Sine: Equation

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

## 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.)

## Examplesin x = 1/2

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} + 1^{2} = 2^{2}.

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.