Brush Up on Laws of Sines and Cosines


So is it really all that important to know or brush up on more advanced mathematics and physics to do carpentry, or even smaller DIY tasks around the house? Well, I guess it depends on just how advanced we’re talking about here. For example, basic trigonometry can be quite helpful in calculating distances and angles that can potentially leave you quite frustrated otherwise. I talked about the Pythagorean Theorem in “Calculators and Carpenters“, but there are a couple other handy math equations you might find useful.

The Law of Cosines

The Law of Cosines is a mathematical law that generalizes the Pythagorean Theorem – in that the Pythagorean Theorem works only for right-angle triangles whereas the Law of Cosines governs all triangles. Law of CosinesTo brush up, the Pythagorean Theorem can be used to find a third leg of a right-angle triangle when 2 of the 3 legs are known, by using the formula c2=a2+b2. The Law of Cosines simply adds another line to this: c2=a2+b2-2ab cos y, where y is the angle opposite c. This law is useful when trying to determine the distance of a third leg of a triangle, when there are no 90 degree angles. Granted, this is probably a rare occurrence for most DIY jobs, but you never know.

The Pythagorean Theorem is usually sufficient for most things, as most structures are vertical – or should be. So if the ground is horizontal and your structure is vertical, you have a right angle and have no need of the Law of Cosines. But it may come handy when say, you’re dealing with a ground-slope and there is no right angle in the equation. In any case, it’s good to know and have in your arsenal. Just remember it as an extension of the Pythagorean Theorem to include non-right-angle triangles.

The Law of Sines

This law is the basis for what’s called triangulation, and is what we use to locate a third point of a triangle when either 2 angles and a side or 2 sides and a non-enclosed angle, are known. You can of course plot your triangle out on a graph with the angles and sides that are known, and complete the triangle by drawing lines based on the known information. For example, in the case where you know 2 sides Determining distance via triangulationand the angle between them, you can draw a line to and from the ends of each of the two known sides completing the triangle.

This is true also for the case where two known angles are connected by a known side. You can simply run the two sides up until they intersect, giving you the third vertex. But this equation comes especially handy when needing to compute problems without the use of graphs, rulers, compasses etc, and when you need accurate, specific numbers – not to mention plotting it on paper will only be as accurate as your eye can tell and the point of your pencil.

  • In the case of the ship in the image above, it would be fairly simple to compute its position based on the two known angles on the beach and the distance l between them. Practically speaking, angles can be measured by two sticks, one pointing in the direction of the ship and the other pointing in the direction of the other known reference point. Both sticks should converge at a preestablished point, and the distance l between the two measured angles needs to be measured. The following equations show how the distance can be calculated using the law of sines:

which is equivalent to

The equation for the law of sines is as follows: a/sinA = b/sinB = c/sinC. It relates the side lengths to the sines of the angles opposite them. For example, a would be the leg that is on the opposite side of the angle A and so on. As an example, let’s take the following known angles and solve the equation: a=10, c=14, and the angle C is 80 degrees. This is a case of having 2 known sides, a and c, and a known non-enclosed angle, C. By the law of sines, we make the following assumption: sinA/10 = sin80°/14.

So if we can figure out the angle A, we can subtract A and C from 180 – as per the triangle postulate – and complete our triangle. Mathematics tells us that sin80°/14 multiplied by the side a, 10, will give us the sinA, 0.703. The inverse of sinA, which is the arcsine, is the angle we’re looking for, the angle A. So, A = arcsine (10sin80°/14) = 44.7°. Now we have the angles A and C, and the sides a and c. By subtracting the sum of angles A and C from 180, you get the angle B, which is 55.3°. Don’t be afraid to use a calculator to solve these kinds of problems ;-), don’t see any shame in that.

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