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.

Answer 1

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 $\boxed{37^\circ}$.