The angle of elevation is the angle above the horizontal that an observer must look to see an object that is higher than the observer. The angle of depression is the angle below the horizontal that an observer must look to see an object that is lower than the observer.
Since both the angle of elevation and angle of depression are the angles from the horizontal line of sight, the angle of elevation is congruent to the angle of depression. The angles are congruent because they are the alternate interior angles.
We can apply what we learned in Target C - What's My Sine? to solve application problems of right triangle trigonometry.
Example 1:
At \(2\) pm the shadow of the Cape Hatteras Lighthouse is \(23.5\) feet long and the angle of elevation is \(83^{\circ}\). Find the height of the lighthouse to the nearest tenth of a foot.
Example 1:
At \(2\) pm the shadow of the Cape Hatteras Lighthouse is \(23.5\) feet long and the angle of elevation is \(83^{\circ}\). Find the height of the lighthouse to the nearest tenth of a foot.
Example 2:
Mr. Pythagoras was walking down the stairs to meet his Math friends and discuss triangles when he accidentally slipped and fell. The stairs make an angle of \(34^{\circ}\) degrees with the floor. If the vertical distance between floors is \(14\) feet, how long was the staircase that he fell down? Round your answer to the nearest tenth of a foot.
Mr. Pythagoras was walking down the stairs to meet his Math friends and discuss triangles when he accidentally slipped and fell. The stairs make an angle of \(34^{\circ}\) degrees with the floor. If the vertical distance between floors is \(14\) feet, how long was the staircase that he fell down? Round your answer to the nearest tenth of a foot.
Solution:
Watch the video for the solution.
Watch the video for the solution.
Example 3:
The angle of elevation from Buster’s eyes to the light fixture in the stairwell is \(47^{\circ}\). If the distance between Buster and the point directly below the light fixture is \(20\) feet, what is the height of the light fixture to the nearest tenth of a foot? (Buster is \(6’ 4”\)). Solution: Watch the video for the solution. |
Example 4
You go fishing with your BFF, Maverick the Hippo, on your fancy new boat. A big storm hits and Maverick is suddenly thrown overboard!! In desperation, you throw a lifesaver to Maverick because for some reason she does not know how to swim. You predict that the angle of depression of the rope is \(35^{\circ}\) degrees and the distance between Maverick and the boat is \(10\) feet. How long does the rope need to be to reach her? Round your answer to the nearest tenth of a foot. If the rope is \(13\) feet long, will it reach Maverick?
You go fishing with your BFF, Maverick the Hippo, on your fancy new boat. A big storm hits and Maverick is suddenly thrown overboard!! In desperation, you throw a lifesaver to Maverick because for some reason she does not know how to swim. You predict that the angle of depression of the rope is \(35^{\circ}\) degrees and the distance between Maverick and the boat is \(10\) feet. How long does the rope need to be to reach her? Round your answer to the nearest tenth of a foot. If the rope is \(13\) feet long, will it reach Maverick?
Whew, the rope is long enough to reach Maverick!
Example 5
The height of a taller office building is \(215\) meters. The angle of depression from the top of the taller building to the top of the shorter building is \(17^{\circ}\) degrees. The height of the shorter building is \(167\) meters. Find the distance between the two office buildings. Solution: \(\begin{align*} \text{tan} (17^{\circ}) &= \frac{28}{x}\\ \frac{x}{1}\cdot \text{tan} (17^{\circ}) &= \frac{28}{x}\cdot\frac{x}{1}\\ \frac{x\cdot \text{tan} (17^{\circ})}{\text{tan} (17^{\circ})} &= \frac{28}{\text{tan} (17^{\circ})}\\ x &= \frac{28}{\text {tan} (17^{\circ})} \approx 91.6 \text {ft} \end{align*}\) |
Quick Check
1) Suppose you are about to enter the tram near the top of Lone Peak at Big Sky Resort in Utah. The angle of depression of your view down the mountain is \(30.8^{\circ}\) degrees. The vertical rise is \(1450\) feet. Determine the distance of the tram ride. Round your answer to the nearest foot.
2) If you are standing on the top of a building that is \(101\) feet tall and looking down at a 2nd building that is \(80\) feet tall with an angle of depression of \(22^{\circ}\). How far away is the 2nd building? Round your answer to the nearest foot.
Quick Check Solutions
1) Suppose you are about to enter the tram near the top of Lone Peak at Big Sky Resort in Utah. The angle of depression of your view down the mountain is \(30.8^{\circ}\) degrees. The vertical rise is \(1450\) feet. Determine the distance of the tram ride. Round your answer to the nearest foot.
2) If you are standing on the top of a building that is \(101\) feet tall and looking down at a 2nd building that is \(80\) feet tall with an angle of depression of \(22^{\circ}\). How far away is the 2nd building? Round your answer to the nearest foot.
Quick Check Solutions