Have you ever gazed up at the stars and wondered how high they are? Or looked down from a hill and tried to guess the distance to the ground below? In our daily lives, we often encounter situations where we’re either looking up at an angle (like admiring a tall skyscraper) or looking down from a height (maybe from your window at a friend below). These are real-life instances of what mathematicians call angles of elevation and angles of depression!
Understanding these angles isn’t just about looking up and down; it’s about diving into the fascinating world of trigonometry. In the word problems that follow, you’ll be exploring various scenarios where you can apply your knowledge to figure out heights, distances, and more. It’s a perfect way to see math in action, and it’s much more engaging when you can relate it to things you see around you. So, sharpen your pencils, grab your protractors, and get ready for some trigonometric adventures!
Word Problems
Mountain Hike
Jake is hiking up a mountain. When he’s 100 meters from the base, he looks up to the peak and measures the angle of elevation to be 30°. How high is the peak above Jake’s line of sight?
Skyscraper Observation
From a point on the ground, Lara observes the top of a skyscraper at an angle of elevation of 45°. If she is 200 meters from the base of the skyscraper, how tall is the skyscraper?
Lighthouse View
A ship is sailing in the sea and is 5 km away from a lighthouse. The captain sees the top of the lighthouse at an angle of elevation of 3°. If the captain’s eye level is 2 meters above the sea level, how tall is the lighthouse?
Bird Watching
Samuel is watching a bird on a tree. The angle of elevation from his eyes to the bird is 20°. If Samuel is standing 30 meters from the tree, how high is the bird from his eye level?
Airplane Descent
An airplane is preparing to land. The pilot notices a building in the distance and measures an angle of depression of 10° to the top of the building. If the airplane is flying at a height of 3000 meters, how far is the airplane from the building?
Pool Dive
Sarah stands at the edge of a diving platform and looks down at the pool below. The angle of depression from her toes to the other end of the pool is 25°. If the pool is 50 meters long, how high is the diving platform?
Observation Wheel
Mark is on an observation wheel and at the highest point. He looks down at his friend and sees him at an angle of depression of 15°. If the observation wheel has a radius of 40 meters, how far is his friend from the base of the wheel?
Hill Descent
A cyclist is about to descend a hill. Before she starts, she measures the angle of depression to the foot of the hill to be 12°. If she is 500 meters from the base of the hill, how high is she currently above the base?
Castle Tower
From a distance of 150 meters, a tourist looks up at a castle’s tower and notices the flag on top at an angle of elevation of 18°. How high is the flag from the tourist’s eye level?
Balloon Adventure
Alice is in a hot air balloon and measures the angle of depression to a landmark on the ground as 26°. If the hot air balloon is 1000 meters above the ground, how far is Alice from the landmark horizontally?
Mountain Peak
Sophia stands 500 meters away from a mountain peak and sees the top at an angle of elevation of 40°. How high is the peak above Sophia’s eye level?
Building Shadow
A tall building casts a shadow 300 meters long. The angle of elevation of the sun is 50°. How tall is the building?
Bridge Observation
John stands under a bridge and looks up to see a bird sitting at an angle of elevation of 60°. If he is 50 meters from the bird horizontally (the horizontal line), how high is the bird above him?
Kite Flying
Emily flies a kite and lets out 100 meters of string at an angle of elevation of 70°. How high is the kite above the ground level?
Canyon Depth
Mark stands at the edge of a canyon and looks down to see a river at an angle of depression of 28°. If he is 400 meters from the river horizontally, how deep is the canyon at that point?
Stadium Lights
A football player looks up at the stadium lights at an angle of elevation of 75°. If he is 20 meters away from the base of the light pole, how high are the lights?
Hilltop Viewing
From a hilltop, Anna observes a car on the ground below at an angle of depression of 12°. If the hill is 500 meters high, how far is the car from the base of the hill?
Rooftop Ladder
A ladder leans against a building, touching the roof at an angle of elevation of 65°. If the ladder is 8 meters long, how high does it reach on the building?
- The Cliff
From a boat at sea, a sailor sees a lighthouse on top of a cliff. The angle of elevation to the lighthouse’s top is 15°. If the sailor is 800 meters from the base of the cliff, how tall is the cliff?
Hot Air Balloon
Jack is in a hot air balloon and measures an angle of depression of 10° to a farmhouse. If the balloon is 500 meters above the ground, how far is the balloon from the farmhouse horizontally?
Answers
57.74 meters
200 meters
261.8 meters
10.92 meters
16952.1 meters
23.31 meters
150.88 meters
106.28 meters
48.74 meters
2190.2 meters
419.55 meters
357.54 meters
86.6 meters
93.97 meters
212.68 meters
74.64 meters
2352.89 meters
7.25 meters
214.32 meters
2834.49 meters