Question

In order to reach the top of a hill, which is 250 feet high, one must travel 2000 feet straight up a road, which leads to the top. Find the number of degrees contained in the angle, which the road makes with the horizontal. Round to the nearest whole degree.

Answer 1

Let $\theta$ be the desired angle in degrees. Using right triangle trigonometry, we see that $\frac{\sin\theta}{250} = \frac{\cos\theta}{2000}$. Cross-multiplying gives $\sin\theta\cdot 2000 = 250 \cos\theta$. Letting $\Delta$ denote the difference, we have $\Delta = \sin\theta - \cos\theta = \cos(90^\circ - \theta) - \cos\theta = -2\sin(45^\circ)\sin\left(45^\circ - \theta\right)$. Hence, $\sin\left(45^\circ - \theta\right) = -\frac{\Delta}{2\sin(45^\circ)}$.

Recall the identities $\sin 2\alpha = 2\sin\alpha\cos\alpha$, $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \sin\beta\cos\alpha$, and $\sin(\alpha - \beta) = \sin\alpha\cos\beta - \sin\beta\cos\alpha$. Applying these identities to $\alpha = 45^\circ$ and $\beta = \theta$, we obtain $2\sin\theta\cos 45^\circ = \Delta$ and $\sin(\theta - 45^\circ) = \frac{\Delta}{2\sin(45^\circ)}$. Equating this value to $-\frac{\Delta}{2\sin(45^\circ)}$, we see that $\theta - 45^\circ = 180^\circ - \theta$. This gives $\theta = 112.5^\circ$ to the nearest $0.5^\circ$, so to the nearest whole number of degrees, our answer is $\boxed{113}$.