Law of Cosines (C)

An extension of the Pythagorean Theorem is the following:

Theorem 800 (Law of Cosines)

Suppose that $\Delta ABC$ is any triangle. Then

$$\vert\vert AB\vert\vert^2 = \vert\vert BC\vert\vert^2 + \ver... ...t\vert BC\vert\vert\cdot\vert\vert AC\vert\vert\cos(\angle BCA)$$

Exercises

1.

A triangle has sides of length 2 and 3 with an included angle of 45 degrees. How long is the other side?

2.

A triangle has sides of length 5 and 3 and the included angle measures $\pi/6$ radians. How long is the remaining side?

3.

A triangle has sides of length 3, 4 and 6. What are the cosines of each of its angles?

4.

A triangle is formed by joining the points (2,0), (9,3) and (5,7). What are the cosines of each of its angles?

5.

A triangle is formed by joining the points (2,3,4), (0,-2,9) and (11,2,0). What are the cosines of it angles?

6.

A triangle is formed by joining the points (0,0,0), (a,b,c) and (x,y,z). Let $\theta$ be the angle whose vertex is (0,0,0). Show that

$$\cos(\theta) = \frac{ax + by + cz}{\sqrt{a^2+b^2+c^2}\sqrt{x^2+y^2+z^2}}$$