Inverse Cosine

Every ratio has its inverse. When you know the adjacent side and the hypotenuse, the cosine links them, and $\arccos$ gives the angle back.

A rope 7 m long is pegged 4 m from the foot of a pole. The 4 m along the ground touches the angle at the peg, so $\cos\left(\theta\right) = \frac{4}{7}$, and $\theta = \arccos\left(\frac{4}{7}\right)$, about 55.15°. That is the angle the rope makes with the ground.

The cosine shrinks as the angle grows: a cosine near 1 means a flat angle, a cosine near 0 an angle near 90°. 4/7 is a little more than a half, and cos 60° is exactly a half, so the angle should be a little under 60°. It is.