Example 6 from Algebra and Trigonometry, 7.2 Right Triangle Trigonometry

Measuring a Distance Indirectly

To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of $57°$ between a line of sight to the top of the tree and the ground, as shown in Figure 13. Find the height of the tree.

A tree with angle of 57 degrees from vantage point. Vantage point is 30 feet from tree.

Figure 13

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We know that the angle of elevation is $57°$ and the adjacent side is 30 ft long. The opposite side is the unknown height.
The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of $57°,$ letting $h$ be the unknown height.

$\begin{matrix} tan θ & = & \frac{opposite}{adjacent} \\ tan (57°) & = & \frac{h}{30} & Solve for h . \\ h & = & 30tan (57°) & Multiply . \\ h & \approx & 46.2 & Use a calculator . \end{matrix}$
The tree is approximately 46 feet tall.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of $\frac{5 π}{12}$ with the ground? Round to the nearest foot.

How did it go?

Solution

Your turn

Solution

Your turn