Question Video: Using the Law of Sines to Calculate an Unknown Angle in a Triangle Mathematics

Video Transcript

𝐴𝐵𝐶 is a triangle, where 𝑎 is equal to nine, 𝑏 is equal to six, and the measure of the angle at 𝐴 is equal to 58.1 degrees. Find the measure of the angle at 𝐵 to the nearest tenth of a degree.

Let’s start by sketching this diagram out. Remember the diagram does not need to be to scale. But it should be roughly in proportion. So we can check the suitability of any answers we get. The measure of the angle at 𝐴 is 58.1 degrees. Remember the side opposite the angle labelled 𝐴 is lowercase 𝑎. So that one is nine. Similarly, the side opposite the angle labelled as 𝐵 is lowercase 𝑏. That’s six.

So we have a non-right-angled triangle, in which we know two sides and the measure of one angle. We need to decide then whether we use the law of cosines or the law of sines. The law of cosines is used when we either know or are trying to calculate the size of the included angle.

In our diagram, the included angle is 𝐶. It’s the one that lies directly between the two sides given. Since we don’t know the size of that angle, we need to use the law of sines: sin 𝐴 over 𝑎 is equal to sin 𝐵 over 𝑏 which is equal to sin 𝐶 over 𝑐. Another thing to look out for our pairs of sides and their associated angles, that is the given angles must sit directly opposite the given side as in this triangle.

With the law of sines, we only need to use two parts of the equation. Here, we know the length of the side 𝑎 and the angle 𝐴. And we know the length of side 𝑏 and we’re trying to calculate the measure of the angle at 𝐵. We use sin 𝐴 over 𝑎 is equal to sin 𝐵 over 𝑏. Let’s substitute what we know into this formula.

Remember we can use this formula either way round. When calculating the angle, it’s sensible to have the sin part of the formula on the top to minimise any rearranging required to solve the equation. In this case, the equation becomes sin of 58.1 degrees over nine is equal to sin of 𝐵 degrees divided by six.

We can solve this equation by multiplying both sides by six. sin 𝐵 is equal to sin of 58.1 degrees divided by nine then multiplied by six. Popping that into our calculator gives us that sin of 𝐵 is equal to 0.5659.

Next, we’ll calculate the inverse sine of both sides of this equation. We do this because the inverse sine of sin 𝐵 is simply 𝐵. We’ll use the exact value we calculated before because this will minimise any errors in calculations from rounding too early. 𝐵 is equal to the inverse sin of 0.5659. This gives us that the measure of the angle at 𝐵 is 34.470.

Correct to the nearest tenth of a degree, the measure of the angle at 𝐵 is 34.5 degrees.