Law of cosines


Law of cosines

From Wikipedia, the free encyclopedia

Figure 1 – A triangle. The angles α,β, and γ are respectively opposite the sides ab, and c.

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) is a statement about a general triangle that relates the lengths of its sides to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states that
c^2 = a^2 + b^2 - 2ab\cos\gamma\ ,
where γ denotes the angle contained between sides of lengths a and b and opposite the side of length c.
The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle γ is a right angle (of measure 90° or π/2 radians), then cos(γ) = 0, and thus the law of cosines reduces to
c^2 = a^2 + b^2 \,
The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known.
By changing which legs of the triangle play the roles of ab, and c in the original formula, one discovers that the following two formulas also state the law of cosines:
a^2 = b^2 + c^2 - 2bc\cos\alpha\,
b^2 = a^2 + c^2 - 2ac\cos\beta\,

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