Billy is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How far is he from his starting point? Express your answer in simplest radical form.
The triangle has sides of 4, 6, and x, where x is the unknown distance. The angle opposite x is 120 degrees, since Billy turned 60 degrees northward from the east. The law of cosines says that $x^2 = 4^2 + 6^2 - 2 \cdot 4 \cdot 6 \cdot \cos 120$. I can simplify this by plugging in the value of $\cos 120 = -\frac{1}{2}$. This gives me $x^2 = 16 + 36 + 24 = 76$. Taking the square root of both sides, I get $x = \sqrt{76}$. I can simplify the radical by factoring out the largest perfect square. This gives me $x = \sqrt{4 \cdot 19} = 2\sqrt{19}$. 