Law of cosines Guide, Meaning , Facts, Information and Description

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, and c be the sides of the triangle and A, B, and C the angles opposite those sides. Then,

This formula is useful for computing the third side of a triangle when two sides' and their enclosed angle's values are known, and in computing the angles of a triangle if all three sides' values are known.

The law of cosines also shows that

iff

The statement cos C = 0 implies that C is a right angle, since a and b are positive. In other words, this is the Pythagorean Theorem and its converse. Although the law of cosines is a broader statement of the Pythagorean Theorem, it isn't a proof of the Pythagorean Theorem, because the law of cosines derivation given below depends on the Pythagorean Theorem.

Table of contents

1 Derivation (for acute angles) 2 Law of cosines using vectors 3 See also 4 External link

Derivation (for acute angles)

Let a, b, and c be the sides of the triangle and A, B, and C the angles opposite those sides. Draw a line from angle B that makes a right angle with the opposite side, b. If the length of that line is x, then which equals

That is, the length of this line is Similarly, the length of the part of b that connects the foot point of the new line and angle C is The remaining length of b is This makes two right triangles, one with legs and hypotenuse c. Therefore, according to the Pythagorean Theorem:

• is always 1, so

Law of cosines using vectors

Using vectors and vector dot products, we can easily prove the law of cosines. If we have a triangle with vertices A, B, and C whose sides are the vectors a, b, and c, we know that:

• since

Using the dot product, we simplify this into

See also

• triangulation
• law of sines

External link

• Several derivations of the Cosine Law, including Euclid's

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