Word Problems
Apply your trigonometry skills to solve real-world problems involving angles of elevation, depression, and practical measurements.
Key Concepts for Word Problems
Angle of Elevation
The angle formed between a horizontal line and the line of sight when looking up at an object. Example: looking up at the top of a building from ground level.
Angle of Depression
The angle formed between a horizontal line and the line of sight when looking down at an object. Example: looking down from a cliff to a boat on the water.
Problem-Solving Strategy
- Read carefully - identify what is given and what you need to find
- Draw a diagram - sketch the situation with a right triangle
- Label everything - mark known values and what you need to find
- Choose your ratio - select sine, cosine, or tangent based on known/unknown sides
- Solve and check - calculate and verify your answer makes sense
Worked Examples
Example 1: Building Height
Problem: From a point 50 meters from the base of a building, the angle of elevation to the top is 62 degrees. How tall is the building?
Show Solution
Step 1: Draw the diagram. The building forms the opposite side, the 50m distance is the adjacent side, and 62 degrees is the angle of elevation.
Step 2: Choose the ratio. We have adjacent and need opposite, so use tangent.
Step 3: Set up and solve.
tan(62) = height / 50
height = 50 * tan(62)
height = 50 * 1.8807
Answer: The building is 94.04 meters tall
Example 2: Finding Distance
Problem: A lifeguard sits in a chair 3 meters above the beach. She spots a swimmer at an angle of depression of 8 degrees. How far is the swimmer from the base of the lifeguard chair?
Show Solution
Step 1: The angle of depression from the lifeguard equals the angle of elevation from the swimmer (alternate interior angles). So we work with an 8-degree angle.
Step 2: The height (3m) is opposite, and we need the adjacent (horizontal distance).
Step 3: Use tangent.
tan(8) = 3 / distance
distance = 3 / tan(8)
distance = 3 / 0.1405
Answer: The swimmer is 21.35 meters from the lifeguard chair
Practice Problems
Problem 1: The Lighthouse
A lighthouse keeper looks down at a ship with an angle of depression of 15 degrees. If the lighthouse is 40 meters tall, how far is the ship from the base of the lighthouse?
Hint
The height (40m) is opposite to the 15-degree angle. Use tan(15) = 40/distance and solve for distance.
Answer
distance = 40 / tan(15) = 40 / 0.2679 = 149.3 meters
Problem 2: The Kite
A kite is flying at the end of a 100-meter string. The string makes an angle of 55 degrees with the ground. How high is the kite above the ground?
Hint
The string is the hypotenuse, and the height is opposite to the 55-degree angle. Use sine.
Answer
height = 100 * sin(55) = 100 * 0.8192 = 81.92 meters
Problem 3: The Skateboard Ramp
A skateboard ramp is 8 feet long and rises to a height of 2.5 feet. What angle does the ramp make with the ground?
Hint
The ramp length is the hypotenuse, and the height is opposite. Use inverse sine.
Answer
angle = sin^(-1)(2.5/8) = sin^(-1)(0.3125) = 18.21 degrees
Problem 4: Two Buildings
From the roof of a 30-meter building, the angle of depression to the base of another building is 35 degrees, and the angle of elevation to the top of that building is 25 degrees. How tall is the second building?
Hint
First find the horizontal distance using the angle of depression and 30m. Then use that distance with the angle of elevation to find the additional height. Add both parts together.
Answer
Distance = 30/tan(35) = 42.84m. Additional height = 42.84 * tan(25) = 19.97m. Total height = 30 + 19.97 = 49.97 meters
Problem 5: The Airplane
An airplane is flying at an altitude of 8,000 feet. The pilot sees the airport runway at an angle of depression of 18 degrees. What is the direct (diagonal) distance from the plane to the runway?
Hint
You need to find the hypotenuse. The altitude (8,000 ft) is opposite to the 18-degree angle. Use sine.
Answer
sin(18) = 8000/distance. distance = 8000/sin(18) = 8000/0.309 = 25,889 feet
Problem 6: The Shadow
A flagpole casts a shadow 28 feet long when the sun's angle of elevation is 52 degrees. How tall is the flagpole?
Hint
The shadow is adjacent to the sun's angle, and the flagpole height is opposite. Use tangent.
Answer
height = 28 * tan(52) = 28 * 1.2799 = 35.84 feet
Problem 7: The Mountain
From a point on the ground, the angle of elevation to the peak of a mountain is 22 degrees. After walking 1,000 meters closer to the mountain, the angle of elevation is 38 degrees. How tall is the mountain?
Hint
Let h = height and d = original distance. Set up two equations: tan(22) = h/d and tan(38) = h/(d-1000). Solve the system.
Answer
From tan(22) = h/d, we get d = h/0.404. From tan(38) = h/(d-1000), we get d-1000 = h/0.781. Substituting: h/0.404 - 1000 = h/0.781. Solving: h = 835 meters
Problem 8: The Cable Car
A cable car travels along a cable from the base of a mountain to a platform at the top. The cable is 450 meters long and makes an angle of 28 degrees with the horizontal. What is the vertical rise of the cable car?
Hint
The cable is the hypotenuse, and the vertical rise is opposite to the 28-degree angle.
Answer
rise = 450 * sin(28) = 450 * 0.4695 = 211.3 meters
Problem 9: The River
To measure the width of a river, a surveyor stands directly across from a tree on the opposite bank. She then walks 80 meters along the riverbank and measures the angle to the tree as 63 degrees. How wide is the river?
Hint
The 80 meters is adjacent to the 63-degree angle, and the river width is opposite.
Answer
width = 80 * tan(63) = 80 * 1.9626 = 157 meters
Problem 10: The Roof
A house has a roof that rises 4 feet for every 12 feet of horizontal distance (run). What is the angle of the roof pitch?
Hint
The rise (4 ft) is opposite and the run (12 ft) is adjacent. Use inverse tangent.
Answer
angle = tan^(-1)(4/12) = tan^(-1)(0.333) = 18.43 degrees
Problem 11: The Zip Line
A zip line connects two platforms. The starting platform is 35 meters high, and the ending platform is 12 meters high. If the horizontal distance between them is 120 meters, what angle does the zip line make with the horizontal?
Hint
The vertical drop is 35 - 12 = 23 meters (opposite). The horizontal distance is 120 meters (adjacent).
Answer
angle = tan^(-1)(23/120) = tan^(-1)(0.1917) = 10.85 degrees
Problem 12: The Staircase
A staircase rises at an angle of 35 degrees. If the horizontal distance covered by the staircase is 4 meters, what is the total length of the staircase (the diagonal distance you walk)?
Hint
The horizontal distance (4m) is adjacent, and you need the hypotenuse. Use cosine.
Answer
cos(35) = 4/length. length = 4/cos(35) = 4/0.8192 = 4.88 meters
Check Your Understanding
- What is the difference between angle of elevation and angle of depression?
- Why is drawing a diagram important for word problems?
- In a problem involving shadows, which trigonometric ratio is most commonly used?
- How do you set up a problem when you need to find the hypotenuse?
Check Your Answers
- Angle of elevation is measured when looking up from horizontal; angle of depression is measured when looking down from horizontal.
- A diagram helps you visualize the right triangle, correctly identify which sides are opposite, adjacent, and hypotenuse, and choose the right trig ratio.
- Tangent is most common because shadows involve the object height (opposite) and shadow length (adjacent).
- Use either sine (if you know the opposite) or cosine (if you know the adjacent), then solve for the hypotenuse by dividing.
Next Steps
- Practice drawing diagrams quickly and accurately
- Look for real-world examples of trigonometry around you
- Move on to Common Mistakes to learn what errors to avoid
- When you feel confident, take the Unit Quiz to test your mastery