"Sine" is math shorthand for the ratio of two sides of a right triangle, expressed as a fraction: The side opposite whatever angle you're measuring is the numerator of the fraction, and the hypotenuse of the right triangle is the denominator. Once you master this concept it becomes a building block for a formula known as the law of sines, which can be used to find missing angles and sides for a triangle as long as you know at least two of its angles and one side, or two sides and one angle.
Recapping the Law of Sines
The law of sines tells you that the ratio of an angle in a triangle to the side opposite it will be the same for all three angles of a triangle. Or, to put it another way:
sin(A)/a = sin(B)/b = sin(C)/c, where A, B and C are the angles of the triangle, and a, b and c are the lengths of the sides opposite those angles.
This form is the most useful for finding missing angles. If you're using the law of sines to find the missing length of a side of the triangle, you can also write it with the sines in the denominator:
a/sin(A) = b/sin(B) = c/sin(C)
Finding a Missing Angle With the Law of Sines
Imagine that you have a triangle with one known angle – let's say angle A measures 30 degrees. You also know the measure of two sides of the triangle: side a, which is opposite angle A, measures 4 units, and side b measures 6 units.
Fill in Known Information
Input all the known information into the first form of the law of sines, which is best for finding missing angles:
sin(30)/4 = sin(B)/6 = sin(C)/c
Next, choose a target; in this case, find the measure of angle B.
Set up the Problem
Setting up the problem is as simple as setting the first and second expressions of this equation equal to each other. No need to worry about the third term right now. So, you have:
sin(30)/4 = sin(B)/6
Find the Known Sine Value
Use a calculator or a chart to find the sine of the known angle. In this case, sin(30) = 0.5, so you have:
(0.5)/4 = sin(B)/6, which simplifies to:
0.125 = sin(B)/6
Isolate the Unknown Angle
Multiply each side of the equation by 6 to isolate the sine measurement of the unknown angle. This gives you:
0.75 = sin(B)
Look up the Unknown Angle
Find the inverse sine or arcsine of the unknown angle, using your calculator or a table. In this case, the inverse sine of 0.75 is approximately 48.6 degrees.
Beware of the ambiguous case of the law of sines, which can arise if you are, as in this problem, given the length of two sides and an angle that is not between them. The ambiguous case is simply a warning that in this specific set of circumstances, there might be two possible answers to choose from. You've already found one possible answer. To parse out another possible answer, subtract the angle you just found from 180 degrees. Add the result to the first known angle you had. If the result is less than 180 degrees, that "result" you just added to the first known angle is a second possible solution.
Finding a Side With the Law of Sines
Imagine that you have a triangle with known angles of 15 and 30 degrees (let's call them A and B respectively), and the length of side a, which is opposite angle A, is 3 units long.
Calculate the Missing Angle
As previously mentioned, the three angles of a triangle always add up to 180 degrees. So if you know two angles already, you can find the measure of the third angle by subtracting the known angles from 180:
180 - 15 - 30 = 135 degrees
So the missing angle is 135 degrees.
Fill in Known Information
Fill the information you already know into the law of sines formula, using the second form (which is easiest when calculating a missing side):
3/sin(15) = b/sin(30) = c/sin(135)
Choose a Target
Choose which missing side you want to find the length of. In this case, for the sake of convenience, find the length of side b.
Set the Problem Up
To set the problem up, you'll choose two of the sine relations given in the law of sines: The one containing your target (side b) and the one you already know all the information for (that's side a and angle A). Set those two sine relations equal to each other:
3/sin(15) = b/sin(30)
Solve for the Target
Now solve for b. Start by using your calculator or a table to find the values of sin(15) and sin(30) and fill them into your equation (for the sake of this example, use the fraction 1/2 instead of 0.5), which gives you:
3/0.2588 = b/(1/2)
Note that your teacher will tell you how far (and if) to round your sine values. They might also ask you to use the exact value of the sine function, which in the case of sin(15) is the very messy (√6 – √2)/4.
Next, simplify both sides of the equation, remembering that dividing by a fraction is the same as multiplying by its inverse:
11.5920 = 2_b_
Switch the sides of the equation for the sake of convenience, since variables are usually listed on the left:
2_b_ = 11.5920
And finally, finish solving for b. In this case, all you have to do is divide both sides of the equation by 2, which gives you:
b = 5.7960
So the missing side of your triangle is 5.7960 units long. You could just as easily use the same procedure to solve for side c, setting its term in the law of sines equal to the term for side a, since you already know that side's full information.