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

Video Transcript

𝐿𝑀𝑁 is a triangle where measure the measure of the angle at 𝐿 is equal to 54 degrees and 30 minutes, the measure of the angle at 𝑁 is equal to 23 degrees and 30 minutes, and 𝑁𝐿 is equal to 16.4 centimetres. Find the lengths of 𝑀𝑁 and 𝐿𝑀, giving the answer to one decimal place.

Let’s begin by sketching a diagram of this triangle. Remember a sketch 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. Here, we have a non-right-angled triangle, for which we know the measure of two of its angles and the length of one of its sides. We can rule out the law of cosines then since that requires a minimum of two known sides.

Instead, we’ll use the law of sines to calculate the missing lengths in this triangle. The law of sines says that 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵 which is equal to 𝑐 over sin 𝐶. Alternatively, that’s often written as sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏 which equals sin 𝐶 over 𝑐.

Remember it’s absolutely fine to use either of these formulae. However, since we’re trying to find missing lengths, we’ll use the first one. This will prevent the amount of rearranging we need to do to solve the equations. The second formula is more suited for finding missing angles.

Next, we’ll label the sides of our triangle. The side opposite the angle 𝐿 is given by lowercase 𝑙, the side opposite the angle 𝑀 is given by lowercase 𝑚, and the side opposite the angle 𝑁 is lowercase 𝑛. At this stage, since we know the length of the side that we’ve called lowercase 𝑚, we’ll need to also work out the measure of the angle at 𝑀.

Remember angles in a triangle add to 180 degrees. So we can calculate the measure of the angle at 𝑀 by subtracting the known angles from 180 degrees. Now, many calculators have a button that will allow us to write degrees in terms of minutes and seconds. If this is the case, then we can pop this straight into our calculator as written. However, if it’s not the case, it’s also useful to know how to change from degrees, minutes, and seconds into decimal form.

Remember minutes and seconds are base 60. So we can write 54 degrees and 30 minutes as 54 and thirty sixtieths of a degree. Thirty sixtieths is the same as 0.5. So 54 degrees and 30 minutes is the same as 54.5 degrees. Similarly, 23 degrees and 30 minutes is equivalent to 23 and a half degrees. We can replace these in our equation to find the measure of the angle at 𝑀. That gives us 180 minus 23.5 plus 54.5, which is equal to 102 degrees.

The very final thing we’ll do before calculating the lengths of 𝑀𝑁 and 𝐿𝑀 is to change the letters in our formula for the law of sins to suit our triangle. It becomes 𝑙 over sin 𝐿 equals 𝑚 over sin 𝑀 which equals 𝑛 over sin 𝑁. Remember we usually only need to use two parts of this at any given time.

If we begin by finding the length of the side 𝑀𝑁, which we called lowercase 𝑙, then we can use these two parts: 𝑙 over sin 𝐿 equals 𝑚 over sin 𝑀. We use these as opposed to 𝑛 over sin 𝑁 because we already know the length of the side labelled with lowercase 𝑚. Substituting the known values into our formula gives us 𝑙 over sin of 54.5 equals 16.4 over sin of 102.

Remember it is absolutely fine to change 54.5 degrees into 54 degrees and 30 seconds as long as we remember to use the appropriate buttons on our calculator. To solve this equation, we’ll multiply both sides by sin of 54.5. That gives us that 𝑙 is equal to 16.4 over sin of 102 multiplied by sin of 54.5. And typing that into our calculator gives us that 𝑙 is equal to 13.649 and so on. Correct to one decimal place, the length 𝑀𝑁 is 13.6 centimetres.

Next, to find the length of 𝐿𝑀, which we called lowercase 𝑛, we’ll use these two parts of the law of sines: 𝑚 over sin 𝑀 equals 𝑛 over sin 𝑁. Substituting what we know into this formula gives us 16.4 over sin of 102 equals 𝑛 over sin of 23.5.

And we can solve this equation by multiplying both sides by sin of 23.5. That gives us 𝑛 is equal to 16.4 over sin of 102 multiplied by sin of 23.5 which is 6.685. Correct to one decimal place, that gives us that the length of 𝐿𝑀 is 6.7 centimetres.