Law of cosines

From Academic Kids

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,

<math>c^2 = a^2 + b^2 - 2ab \cos C . \;<math>

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

The law of cosines also shows that

<math>c^2 = a^2 + b^2\;<math>   if and only if   <math>\cos C = 0 . \;<math>

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.

Contents

Proof

Vector triangle

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:

<math>\mathbf{a = b - c} \;<math>

and

<math>\mathbf{(b - c)\cdot (b - c) = b\cdot b - 2 b\cdot c + c\cdot c}. \;<math>

Using the dot product, we simplify the above into

<math>\mathbf{|a|^2 = |b|^2 + |c|^2 - 2 |b||c|}\cos \theta. \;<math>


Alternate proof (for acute angles)

Missing image
Law_of_cosines_proof.png
Triangle

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 sin C = x/a, which implies x = a sin C.

That is, the length of this line is a sin C. Similarly, 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 ba cos C. This makes two right triangles, one with legs a sin C and ba cos C and hypotenuse c. Therefore, according to the Pythagorean theorem:

<math>c^2 = (a \sin C)^2 + (b - a \cos C)^2\,<math>
<math>= a^2 \sin^2 C + b^2 - 2 ab \cos C + a^2 \cos^2 C\,<math>
<math>= a^2 (\sin^2 C + \cos^2 C) + b^2 - 2ab \cos C\,<math>
<math>=a^2+b^2-2ab\cos C\,<math>

because

<math>\sin^2 C + \cos^2 C=1.\,<math>

Transposing

By transposing

<math>c^2=a^2+b^2-2ab\cos C,\,<math>

we can find C:

<math>cos C= \frac{a^2+b^2-c^2}{2ab}<math>

See also

External link

de:Kosinussatz es:Teorema del coseno fr:Thorme d'Al-Kashi ko:코사인 법칙 it:Legge del coseno he:משפט הקוסינוסים nl:Cosinusregel ja:余弦定理 pl:Twierdzenie cosinusw ru:Теорема косинусов sl:Kosinusni izrek uk:Теорема косинусів zh:余弦定理

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