The Law of Sines is a relationship between side lengths of a triangle and the sine of the opposite angle. It is derived from information we get trying to find the height of a triangle relative to one of the sides being called the base.
The Law of Cosines is used to find a third side length when we have two side lengths and the measure of the angle that lies between them, referred to in geometry as SAS or side-angle-side. At 90°, the cosine is zero and the extra term - 2abcosC becomes 0 and we get the Pythagorean Theorem.
Besides SAS, the Law of Cosines is useful if we know the all three sides (SSS) and want to find the measure of the angles. By rearranging the variable with algebraic manipulation we get
cosA = (b² + c² - a²)/2bc
cosB = (a² + c² - b²)/2ac
cosC = (a² + b² - c²)/2ab
The numerator might be 0 or less, but that is not a problem since cosine can take any value from -1 to 1, including 0 when the angle is 90°. Since no side of a legitimate can be of length 0, we do not have to worry about the denominator being 0, so all the possible examples we have of triangles will give use three cosine values, and we can use the inverse cosine function to find the three angles.
Besides SAS, the Law of Cosines is useful if we know the all three sides (SSS) and want to find the measure of the angles. By rearranging the variable with algebraic manipulation we get
cosA = (b² + c² - a²)/2bc
cosB = (a² + c² - b²)/2ac
cosC = (a² + b² - c²)/2ab
The numerator might be 0 or less, but that is not a problem since cosine can take any value from -1 to 1, including 0 when the angle is 90°. Since no side of a legitimate can be of length 0, we do not have to worry about the denominator being 0, so all the possible examples we have of triangles will give use three cosine values, and we can use the inverse cosine function to find the three angles.
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