Law of cosines
In trigonometry, the law of cosines is a statement about arbitrary triangles which generalizes the Pythagorean theorem by correcting it with a term proportional to the cosine of the opposing angle. Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. Then
This formula is useful to compute the third side of a triangle when two sides and the enclosed angle are known, and to compute the angles of a triangle if all three sides are known.
The law of cosines also shows that
iff cos C = 0 (since a, b > 0), which is equivalent to C being a right angle. (In other words, this is the Pythagorean Theorem and its converse.)
Derivation (for acute angles)
Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. Draw a line from angle B that makes a right angle with the opposite side, b. The length of this line is a sin C, and the length of the part of b that connects the foot point of the new line and angle C is a cos C. The remaining length of b is b - a cos C. This makes two right triangles, one with legs a sin C, b - a cos C and hypotenuse c. Therefore, according to the Pythagorean Theorem:
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Law of cosines using vectors
Using vectors and the dot product, we can easily prove the law of cosines. If we have a triangle with the edgepoints A, B and C which sides are the vectors a, b and c, we know that:
Since
Using the dot product, we simplyfy this into:
See also
External link
Referenced By
Law of tangents | List of mathematical topics (J-L) | List of trigonometry topics
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