Sine Law Examples

The sine law is a proportion that compares the ratio of each side of a triangle to it's included angle. If you don't understand this previous sentence, don't worry! No need to head back to the help pages on ratios! Just think of the sine law as a formula. The sine law is shown below:

where a, b, and c are the three sides of the triangle AND A, B, and C are the three angles, as shown in the diagram below:

You know you are able to use the sine law when the question gives you three things:

• The size of an angle
• The length of a side OPPOSITE that angle
• ANY other piece of information.

For example, you CAN use the sine law to solve the triangle below:

But you CANNOT use the sine law to solve this triangle:

With that in mind, let's look at some examples:

Example 1 - Finding the length of a side using the sine law.

Find the length of side c in the following triangle:

Solution:

First, write out the sine law, and then fill in the information that is given in the question:

Notice that in our 'filled in' version above, we have a ratio that has no information associated with it (a and A). We will just ignore that ratio for now (as we don't need either of those values) and we will focus on the other two ratios. Notice the ratio that uses side b and angle B is 'complete' - that is to say we have values for both the numerator and denominator of the fraction. You NEED a complete ratio in the sine law in order to use the sine law. Let's write the two ratios we are going to use again:

Multiply both sides by sin 20° to isolate c, which leaves us with:

Type the left hand side into your calculator (make sure your calculator is in degree mode) and you should arrive at an answer of:

Which we can round to one decimal place to arrive at an answer of 9.6 mm for side c.

Example 2 - Finding an angle using the sine law

Find the length of angle A in the following triangle:

Solution:

Again, we know we can use the sine law because we are given an angle and the side opposite that angle and one other piece of information.

Write out the sine law and fill in the given information:

We will focus on the 'complete fraction' and the ratio that contains angle A. Ignoring the middle ratio, we have:

To solve for A, you can use the 'cross multiply and divide' technique. This technique leaves you with the following equation:

and then:

Don't forget, in order to use your calculator to solve for an angle, you must use the 2nd function of sin. Common keypresses are:

D.A.L. Calculators: .4438078
and:
Non D.A.L. Calculators: .4438078

You should arrive at an answer of A = 26.3°.

For more information, consult your notes and your teacher. The above two examples show the only two possible uses of the sine law (solving for a side and solving for an angle). Questions you see may be more complex, but they all can be solved starting with the skills you have seen here. Now you may try a couple of sine law questions, or head back to the sine law main page.