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* An altitude is a line segment that has one endpoint at a vertex of a triangle intersects the opposite side at a right angle. See triangles.
This labeling scheme is comßmonly used for non-right triangles. Capital letters are angles and the corresponding lower-case letters go with the side opposite the angle: side a (with length of a units) is across from angle A (with a measure of A degrees or radians), and so on.
The sine equations are
We can rearrange those by solving each for x (multiply by c on both sides of the left equation, and by a on both sides of the right):
Now the transitive property says that if both c·sin(A) and a·sin(C) are equal to x, then they must be equal to each other:
We usually divide both sides by ac to get the easy-to-remember expression of the law of sines:
We could do the same derivation with the other two altitudes, drawn from angles A and C to come up with similar relations for the other angle pairs. We call these together the law of sines. It's in the green box below.
The law of sines can be used to find the measure of an angle or a side of a non-right triangle if we know:
Use the law of sines to find the missing measurements of the triangles in these examples. In the first, two angles and a side are known. In the second two sides and an angle. Notice that we need to know at least one angle-opposite side pair for the Law of Sines to work.
First use the Pythagorean theorem to derive two equations for each of the right triangles:
Notice that each contains and x2, so we can eliminate x2 between the two using the transitive property:
Then expand the binomial (b - y)2 to get the equation below, and note that the y2 cancel:
Now we still have a y hanging around, but we can get rid of it using the cosine solution, notice that
Substituting c·cos(A) for y, we get
which is the law of cosines
The law of cosines can be used to find the measure of an angle or a side of a non-right triangle if we know:
We could again do the same derivation using the other two altitudes of our triangle, to yield three versions of the law of cosines for any triangle. They are listed below.
The Law of Cosines is just the Pythagorean relationship with a correction factor, e.g. -2bc·cos(A), to account for the fact that the triangle is not a right triangle. We can write three versions of the LOC, one for every angle/opposite side pair:
Use the law of cosines to find the missing measurements of the triangles in these two examples. In the first, the measures of two sides and the included angle (the angle between them) are known. In the second, three sides are known.
(Hint: Always use the LOS first if you can. It's simpler. When that fails, use the LOC, but then you can usually use other means to fill in the rest of the measurements.)
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