Video: Using the Law of Sines to Calculate Unknown Lengths in a Triangle

Video Transcript

𝐴𝐵𝐶 is an obtuse-angled triangle at 𝐴, where 𝑏 is equal to 15 centimetres, tan 𝐶 is equal to six-fifths, and the measure of the angle at 𝐵 is 27 degrees. Find lengths 𝑎 and 𝑐, giving the answer to the nearest integer.

It’s always sensible to begin by sketching a diagram out. This 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. Remember the angle at 𝐴 is obtuse. So this needs to be apparent on the diagram. Since we know that tan 𝐶 is equal to six-fifths, we can calculate the measure of the angle at 𝐶 by solving this equation.

Since inverse tan of tan 𝐶 is simply 𝐶, we find the inverse tan of both sides. 𝐶 is equal to the inverse tan of six-fifths. The measure of the angle at 𝐶 then is 50.194 and so on. Wherever possible, we should try to use this exact value for the measure of the angle at 𝐶 to prevent any errors by rounding too early.

Next, we can work out the measure of the angle at 𝐴. Angles in a triangle add to 180 degrees. So we can subtract the angles we know from 180. 180 minus 27 plus 50.194 — that’s the measure of the angle at 𝐶 — is equal to 102.805 and so on. That’s obtuse as required.

So we have a non-right-angle triangle, for which we know the length of one of the sides, but all of the angles. This means we can use the law of sines to find any missing measurements. We don’t use the law of cosines as that requires a minimum of two known sides.

The formula for the law of sines is 𝑎 over sin 𝐴 which equals 𝑏 over sin 𝐵 which equals 𝑐 over sin 𝐶. This can be alternatively written as sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏 which equals sin 𝐶 over 𝑐. Since we’re trying to find the measurements of the lengths in this triangle, we’ll use the first form.

It doesn’t really matter which form we choose as we’ll get the right answer either way. But by choosing the first form when we’re trying to find the lengths of the triangle, we’ll reduce the need for too much rearranging. Let’s label the sides of our triangle.

The side opposite the angle 𝐴 is denoted by lowercase 𝑎, the side opposite angle 𝐵 is lowercase 𝑏, and the side opposite angle 𝐶 is lowercase 𝑐. If we choose first to calculate the length of the side 𝑎, we only need to use the two parts of the formula 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵. At this point, we’re not interested in the angle at 𝐶 or the side 𝑐.

Substituting the relevant values from the triangle into this formula gives us 𝑎 over sin of 102.8 equals 15 over sin of 27. Once again, we’ll use the exact value if we can for the measure of the angle at 𝐴. To solve this equation, we can multiply both sides by sin of 102.8. That gives us 𝑎 is equal to 15 over sin of 27 multiplied by sin of 102.8. Popping that value into our calculator gives us 32.2185. Correct to the nearest integer, 𝑎 must be 32 centimetres.

We can follow the same process to calculate the length of the side 𝑐. Since we’re no longer interested in the angle at 𝐴 or the side 𝑎, we use 𝑏 over sin 𝐵 equals 𝑐 over sin 𝐶. Substituting these values into our formula gives us 15 over sin of 27 is equal to 𝑐 over sin of 50.194.

Again, we’ll use the exact value for the measure of the angle at 𝐶. We solve this equation by multiplying both sides by sin of 50.194. 𝑐 is, therefore, equal to 15 over sin of 27 multiplied by sin of 50.194, which if we pop into our calculator gives us a value of 25.382.

Correct to the nearest integer, that’s 𝑐 is equal to 25 centimetres.