Video: Using the Law of Sines to Find an Unknown Length in Terms of the Sine Function

Video Transcript

𝐴𝐵𝐶 is a triangle where 𝑎 is 96 and the measure of the angle at 𝐵 is equal to three times the measure of the angle at 𝐴, which is equal to 90 degrees. Find length 𝑐 giving the answer in terms of sine.

It’s always sensible to begin by sketching a diagram out. It doesn’t need to be to scale, but it should be roughly in proportion, so we can check the suitability of any answers we get. We’re told that the measure of the angle at 𝐵 is equal to three times the measure of the angle at 𝐴 and that these two expressions are equal to 90 degrees. This of course means that the measure of the angle at 𝐵 is 90 degrees. It’s a right angle.

Let’s look at 𝐴 then. Since three times the measure of the angle 𝐴 is 90 degrees, we can work out the measure of the angle at 𝐴 by dividing through by three. And doing so, we can see that the measure of the angle at 𝐴 is 30 degrees. We can also work out the measure of the angle at 𝐶. Angles in a triangle sum to 180 degrees. So we can subtract 90 and 30 from 180. And that tells us the measure of the angle at 𝐶 is 60 degrees.

Now, let’s label the sides of the triangle. The side opposite angle 𝐴 is lowercase 𝑎. The side opposite angle 𝐵 is lowercase 𝑏. And the side opposite angle 𝐶 is lowercase 𝑐. We want to calculate the length of 𝑐. Now normally, we could use right angle trigonometry here. However, the question has asked us to calculate it in terms of sine.

Instead, we’ll use the law of sines that’s 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, which equals 𝑐 over sin 𝐶. Or alternatively, sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏, which equals sin 𝐶 over 𝑐. Since we’re trying to calculate the length of a side, we’ll use the first form. In fact, we can actually use either form of this formula. However, we’ll need to do less rearranging if we use the first one this time.

Remember, we usually only need to use two parts of this formula. Here, we know the length of the side 𝑎 and we’re trying to find an expression for the side marked 𝑐. We’re going to use 𝑎 over sin 𝐴 and 𝑐 over sin 𝐶 then.

Let’s substitute the values from our triangle into the formula. We get 96 over sin 30 is equal to 𝑐 over sin 60. We want to make 𝑐 the subject. So we’ll multiply both sides of this equation by sin 60. And that gives us 𝑐 is equal to 96 over sin 30, all multiplied by sin 60. Remember, we can write sin 60 as sin 60 over one. And then, when we multiply the numerator of the first fraction by the numerator of the second fraction, we get 96 sin 60. And multiplying the two denominators, we get sin 30.

We were told to leave our answer in terms of sine. So we’ve finished. 𝑐 is equal to 96 sin 60 over sin 30.