The Law of Sines tells us that the ratio between the sine of an angle and the side opposite will be constant for any angle in a triangle.
If we have at least two sides and an angle or two angles and a side, we can use it to find the missing values.
Consider a triangle with angles , and and sides , and , where is the side opposite , opposite and opposite .
If , and , solve the triangle.
We can start by finding using the Law of Sines:
Since we know that
We can calculate
Since dividing by 1 equals the numerator, we know:
Then use to find
We can now find since we know that all the angles in a triangle add up to
Now to find , we can use the Law of Sines again:
Calculate the known values:
Multiply both sides by A:
Now that we have the missing values, we can use the Law of Sines to check that all ratios are equal:
See also Law Of Cosines.