Wednesday, January 20, 2010
Wednesday Math, Vol. 106: The Law Of Sines
Last week, the math post was about the Law of Cosines, so naturally this week is about the Law of Sines. The two names sound like the results should be very similar, but they are stated in very different ways. The Law of Cosines is the generalization of the Pythagorean Theorem, while the Law of Sines is about the relationships between side lengths and the sine function of the opposite angle in a triangle.
You may recall that the simplest formula for the area of a triangle is ½bh, one half the base times the height. If you only have the side lengths and don't know the height, it can be found as a formula involving the sine function of either the angle at the left side of the base or the angle at the right side of the base. Since h = a × sin C and h = c × sin A, we can set the two formulas equal to each other and get a × sin C = c × sin A, and by dividing we can get a ratio a/sin A = c/sin C. Since the choice of base is arbitrary and we could do this with any two angles in the triangle, we also get a/sin A = b/sinB = c/sin C.
Wandering around the web, I also found this proof on a Japanese website which involves inscribing a triangle inside a circle. Instead of finding the height of the triangle, this proof uses the fact that the measure of angle A is the same as any inscribed angle that also has lines that cross the circle at points B and C. The triangle with thin lines is a right triangle with hypotenuse equal to 2R, twice the radius, which could also be called D, the diameter. The definition of sine is a ratio between sides, so sin A = a/2R, and with a little algebraic manipulation we get 2R = a/sinA. We could do the same for sides b and c, and get the statement of the Law of Sines shown in the second illustration.
Next week, a little more about triangles and circles, shapes that have many surprising connections.