How to find if view of sun will be obstructed? I know the following:


*

*My elevation ($4,000$ ft.)

*The mountain's elevation ($7,200$ ft.)

*The distance from me to the mountain ($2$ miles)

*The solar elevation ($38$)


How can I tell if the sun will be behind the mountain?

Here is a picture that attempts to describe the situation:

The sun will be obscured by the peak if the angle of elevation to the peak $\beta$ is larger than the angle of elevation to the sun, which we know is $38^{\circ}$.  Otherwise, the sun will not be obscured.
We are given that the distance from You to $A$ is 2 miles, and that the difference in altitude from you to the peak is $7200-4000 = 3200$ feet.  Hence
$$ \tan(\beta) = \frac{\text{opp}}{\text{hyp}} = \frac{3200\text{ ft}}{2\text{ mi}} = \frac{3200\text{ ft}}{2\text{ mi}} \cdot \frac{1\text{ mi}}{5280\text{ ft}} \approx 0.30 $$
(note that the units cancel each other out---the tangent is a unitless quantity).  Then
$$ \beta \approx \arctan(0.30) \approx 17^\circ.$$
For those familiar with the unit circle and the difficulties of defining inverses of periodic functions, note that we are working exclusively in the first quadrant, thus the angle we want really is $\arctan(0.30)$, rather than this plus some integer multiple of $90^{\circ}$.
Therefore, since the angle to the peak (approx. $17^{\circ}$) is less than the angle to the sun ($38^{\circ}$), the peak will not obstruct the sun.
