Trigonometry has many applications in science and engineering. In this section we will present just a few examples from surveying and navigation. The angle made by the line of sight of an observer on the ground to a point above the horizontal is called the angle of elevation. In Figure \(\PageIndex\), \(\angle BAC\) is the angle of elevation.
At a point 50 feet from a tree the angle of elevation of the top of the tree is \(43^\). Find the height of the tree to the nearest tenth of a foot. Solution Let \(x=\) height of tree. \(\begin <\tan 43^> & = & > \\ & = & > \\ & = & (50)> \\ & = & \\ & = & \end\) Answer: \(x = 46.6\) feet.
The angle made by the line of sight of an observer above to a point on the ground is called the angle of depression. In FIgure \(\PageIndex\), \(\angle ABD\) is the angle of depression.
From an airplane 5000 feet above the ground the angle of depression of an airport is \(5^\). How far away is the airport to the nearest hundred feet? Solution Let \(x =\) distance to airport. \(\angle ABC = 85^\). \(\begin <\cos 85^> & = & > \\ & = & > \\ & = & \\ & = & = 57,300> \end\) Answer: 57,300 feet
A road rises 30 feet in a horizontal distance of 300 feet, Find to the nearest degree the angle the road makes with the horizontal. Solution \(\begin & = & > \\ & = & \\ & = & > \end\) Answer: \(6^\).
1. At a point 60 feet from a tree the angle of elevation of the top of the tree is \(40^\). Find the height of the tree to the nearest tenth of a foot. 2. At a point 100 feet from a tall building the angle of elevation of the top of the building is \(65^\). Find the height of the building to the nearest foot. 3. From a helicopter 1000 feet above the ground the angle of depression of a helinort is \(10^\). How far away is the heliport to the nearest foot? 4. From the top of a 100 foot lighthouse the angle of depression of a boat is \(15^\). How far is the boat from the bottom of the lighthouse (nearest foot)? 
5. A road rises 10 feet in a horizontal distance of 400 feet. Find to the nearest degree the angle the road makes with the horizontal. 6. If a 20 foot telephone pole casts a shadow of 43 feet , what is the angle of elevation of the sun? 
7. A 20 foot ladder is leaning against a wall, It makes an angle of \(70^\) with the ground. How high is the top of the ladder from the ground (nearest tenth of a foot)? 8. The angle of elevation of the top of a mountain from a point 20 miles away is \(6^\). How high is the mountain (nearest tenth of a mile)?
This page titled 5.3: Applications of Trigonometry is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Henry Africk (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.
