Trigonometric Ratios Application

Problem 202: Suppose we are looking out a window that is 50 ft from the ground and we look down at a $45^{\circ}$ angle (this is called the angle of depression) to spot a fire on the ground. Approximately how far away from us in is the fire? Why is this only an approximation?

Solution: The angle between the person looking down the window and the high of the building is $45^{\circ}$ . Using this information, we can build a right-triangle where one of the side is the high of the building (50 feet) and the hypothenuse we will call it $x$ which is what we are looking for. Then

$\begin{align*} \cos (45^{\circ}) & = \frac{50}{x} \\ x & = \frac{50}{\cos(45^{\circ})} \\ & = \frac{50}{\frac{\sqrt{2}}{2}} \\ & = \frac{100}{\sqrt{2}} \\ & = \frac{100}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \\ & = \frac{100\sqrt{2}}{2} \\ & = 50\sqrt{2}. \end{align*}$

Hence, the fire is approximately $50\sqrt{2}$ feet away from us. This is only an approximation because we don’t know how fast the fire is spreading.

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