Let's illustrate the angle, θ, that corresponds to a sine value of 12.
Examining the diagram, we see a right triangle with a hypotenuse of 1 and where the opposite leg to θ is half the length of the hypotenuse. This fits the description of a 30^(∘)-60^(∘)-90^(∘) triangle where θ is 30^(∘).
When we know θ we have to find the corresponding radian. Half a lap around the unit circle corresponds to an angle of 180^(∘) and an arc length of π. With this information, we can write the following equation.
180^(∘)=π
Since 30^(∘)(6)=180^(∘), we can determine the number of radians that corresponds to a 30^(∘) angle if we divide both sides by 6.
An angle of 30^(∘) corresponds to a radian of π6. This is one of the equation's solutions.
We find an additional solution on the opposite side of the y-axis. If we subtract the reference angle from π we can find the angle.
θ_2=π-π/6=5π/6
Let's illustrate the second solution.
A third and fourth solution we can obtain by adding 2π to the first and second solution.
Notice that we can add an infinite number of laps n around the unit circle, which will give us an infinite number of solutions. With this information, we can write the following solution sets.
π/6+2π(n) and 5π/6+2π(n)
Let's show the four first solutions on a sine curve.
