Roger fastens a rope to the ground that extends from a pole that has been placed in the ground. The rope is 8 feet long, and the pole is 5 feet tall. Using the inverse of sine, what is the angle formed where the rope is fastened to the ground? Round your answer to the nearest whole degree.
We can create a right triangle with the rope as the hypotenuse and the pole and the ground as the legs of the triangle.
Let $x$ be the angle formed between the rope and the ground.
We have the side opposite $x$ as 5 feet and the hypotenuse as 8 feet.
Using the definition of sine, we have $\sin x = \frac{5}{8}$.
Taking the inverse sine of both sides of the equation, we have $\arcsin\left(\sin x\right) = \arcsin\left(\frac{5}{8}\right)$.
Since $-\frac{\pi}{2} \leq \arcsin x \leq \frac{\pi}{2}$, we know that $x$ is between $-90^\circ$ and $90^\circ$.
Using a calculator or a table of trigonometric values, we find that $x \approx 36.87^\circ$.
Rounded to the nearest whole degree, the angle is $\boxed{37^\circ}$.