i want to understand the law of cosines

Question

with a right triangle it is easy to understand which side is a, b, or c,

so that a^2 + b^2 = c^2

with an acute or obtuse triangle everything seems more confusing.

which side is the hypotenuse, etc?

is the cosine still defined as adjacent over the hypotenuse?

on the internet no one shows solutions using a/b, a/c, etc.

only square roots and approximations.

thanks

Answer 1

In a right triangle, the side opposite to the right angle is the hypotenuse. If the triangle is not a right triangle, then there is no "hypotenuse". However, the sides and angles of the triangle are related by the Law of Cosines. If the sides are denoted by a, b, and c, and if the angles are denoted by A, B, and C, where A is the angle opposite side a, B is the angle opposite side b, and C is the angle opposite side C, then:

a² = b² + c² - 2bccosA

b² = a² + c² - 2accosB

c² = a² + b² - 2abcosC

Note that if one of the angles of the triangle is a right angle (say angle C), then the version of the Law of Cosines that involves angle C reduces to the Pythagorean Theorem. So, the Pythagorean Theorem is a special case of the Law of Cosines.