Law of cosines & dot product
Jan 30, 2023
Trigonometry
The law of cosines is a generalized theorem of the pythagorean theorem. It relates the lengths of the sides of a triangle to the cosine of one of its angle.
When given a triangle ABC, the law of cosines goes like this:
The law of cosines generalizes the pythagorean theorem because the pythagorean theorem holds only for right triangles. In this case,
cos(θ) becomes 0, since θ is 90°. The formula is abbreviated to:
Using the cosine law, we can derive the dot product of vectors.
Using the law of cosines,
The dot product has this property:
So,
It can be shown like this.
