Question

Find the angle of elevation of the sun from the ground to the top of a tree when a tree that is 10 yards tall casts a shadow 14 yards long. Round to the nearest degree.

Answer 1

Approximately
`36^(\circ)`, assuming that the tree is upright.

Explanation:

Refer to the diagram attached. The upright tree and its shadow are in a right triangle. In this right triangle, the angle of elevation of the sun would be the angle adjacent to the shadow of this tree. At the same time, this angle would be opposite to the side representing the tree. Thus, the tangent of this angle could be represented as:

`\begin{aligned}\tan(\text{angle of elevation}) &= \frac{\text{opposite}}{\text{adjacent}} \\ &= \frac{10\; \text{yard}}{14\; \text{yard}} \\ &= (5)/(7)\end{aligned}`.

Take the arctangent of this value to find the measure of this angle:

`\begin{aligned} (\text{angle of elevation}) &= \arctan(\tan(\text{angle of elevation})) \\ &= \arctan\left((5)/(7)\right) \\ &\approx 0.620 * (360^(\circ))/(2\, \pi) \\ &\approx 36^(\circ)\end{aligned}`.

Thus, the angle of elevation of the sun at this moment would be approximately
`36^(\circ)`.