Word Problems
Apply trigonometric concepts to solve real-world problems involving angles, distances, heights, and periodic phenomena.
Learn
Trigonometry has countless real-world applications. This lesson focuses on translating word problems into mathematical models using trigonometric functions.
Common Types of Trigonometry Word Problems
1. Angle of Elevation and Depression
- Angle of elevation: The angle measured upward from the horizontal line of sight to an object above
- Angle of depression: The angle measured downward from the horizontal line of sight to an object below
- Both create right triangles that can be solved with SOH-CAH-TOA
2. Navigation and Bearings
- Bearings are measured clockwise from north (0 degrees to 360 degrees)
- Often involves finding distances or directions using the Law of Sines or Cosines
3. Periodic Phenomena
- Tides, temperatures, daylight hours, and sound waves follow sinusoidal patterns
- Model with equations like y = A sin(Bx + C) + D or y = A cos(Bx + C) + D
4. Triangle Applications
- Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
- Law of Cosines: c^2 = a^2 + b^2 - 2ab cos(C)
- Use when you don't have a right triangle
Problem-Solving Framework
- Read carefully: Identify what is given and what you need to find
- Draw a diagram: Sketch the situation and label known values
- Choose the right tool: SOH-CAH-TOA, Law of Sines, Law of Cosines, or sinusoidal model
- Set up and solve: Write the equation and solve for the unknown
- Check your answer: Does it make sense in context?
Examples
Study these worked examples to see how to approach different types of problems.
Example 1: Angle of Elevation
Problem: A surveyor stands 100 meters from the base of a building. The angle of elevation to the top of the building is 32 degrees. How tall is the building?
Solution:
- Draw a right triangle with the building as the vertical side
- The horizontal distance (adjacent) is 100 m; we need the height (opposite)
- Use tan: tan(32 degrees) = height / 100
- height = 100 * tan(32 degrees) = 100 * 0.6249 = 62.49 meters
Answer: The building is approximately 62.5 meters tall.
Example 2: Angle of Depression
Problem: From a lighthouse 50 meters above sea level, a boat is spotted at an angle of depression of 15 degrees. How far is the boat from the base of the lighthouse?
Solution:
- The angle of depression equals the angle of elevation from the boat
- We have the opposite side (50 m) and need the adjacent side (distance)
- Use tan: tan(15 degrees) = 50 / distance
- distance = 50 / tan(15 degrees) = 50 / 0.2679 = 186.6 meters
Answer: The boat is approximately 186.6 meters from the lighthouse.
Example 3: Law of Cosines
Problem: Two ships leave port at the same time. Ship A travels at 20 km/h on a bearing of 045 degrees. Ship B travels at 25 km/h on a bearing of 120 degrees. How far apart are they after 2 hours?
Solution:
- After 2 hours: Ship A traveled 40 km, Ship B traveled 50 km
- The angle between their paths: 120 - 45 = 75 degrees
- Use Law of Cosines: c^2 = 40^2 + 50^2 - 2(40)(50)cos(75 degrees)
- c^2 = 1600 + 2500 - 4000(0.2588) = 4100 - 1035.2 = 3064.8
- c = sqrt(3064.8) = 55.4 km
Answer: The ships are approximately 55.4 km apart.
Practice
Solve these word problems. Draw a diagram for each one before calculating.
Problem 1: A ladder leans against a wall, making a 70-degree angle with the ground. If the foot of the ladder is 4 feet from the wall, how long is the ladder?
Show Hint
Use cosine: the adjacent side is 4 feet, and you need the hypotenuse (ladder length).
Problem 2: An airplane takes off at a 12-degree angle of elevation. After flying 2 miles, what is the airplane's altitude?
Show Hint
Use sine: you have the hypotenuse (2 miles) and need the opposite side (altitude).
Problem 3: From the top of a 200-foot cliff, the angles of depression to two boats in a line directly out to sea are 35 degrees and 20 degrees. How far apart are the boats?
Show Hint
Find the horizontal distance to each boat separately, then subtract.
Problem 4: A triangular plot of land has sides of 120 m, 150 m, and 100 m. Find the largest angle of the triangle.
Show Hint
The largest angle is opposite the longest side. Use the Law of Cosines.
Problem 5: A Ferris wheel has a diameter of 50 meters and its center is 30 meters above the ground. If it takes 3 minutes to complete one revolution, write an equation for your height h(t) above the ground after t minutes, starting from the lowest point.
Show Hint
Use a cosine function with amplitude 25 (radius), period 3 minutes, and vertical shift 30. Start with -cos since you begin at the bottom.
Problem 6: Two fire towers are 30 km apart. Tower A spots a fire at a bearing of N40 degreesE. Tower B spots the same fire at a bearing of N70 degreesW. How far is the fire from Tower A?
Show Hint
Draw the triangle and find all angles. Then use the Law of Sines.
Problem 7: A ramp needs to rise 3 feet over a horizontal distance of 20 feet. What angle does the ramp make with the ground?
Show Hint
Use arctan (inverse tangent) with opposite = 3 and adjacent = 20.
Problem 8: The height of the tide at a beach can be modeled by h(t) = 4sin(pi*t/6) + 6, where h is in feet and t is in hours after midnight. What is the maximum height of the tide, and when does it first occur?
Show Hint
Maximum of sine is 1. For when it occurs, set the argument of sine equal to pi/2.
Problem 9: A pendulum swings through an arc length of 16 cm. If the pendulum is 40 cm long, through what angle (in radians) does it swing?
Show Hint
Use the arc length formula: s = r*theta, where s is arc length and r is radius.
Problem 10: A surveyor needs to find the distance across a lake. From point A, she measures the distance to point B (along the shore) as 500 m. The angle at A to point C (across the lake) is 68 degrees, and the angle at B to point C is 54 degrees. Find the distance AC.
Show Hint
Find angle C first (angles in a triangle sum to 180 degrees), then use Law of Sines.
Check Your Understanding
Answer these questions to test your comprehension.
Question 1: When would you use the Law of Cosines instead of SOH-CAH-TOA?
Question 2: A tree casts a shadow 25 meters long when the angle of elevation of the sun is 53 degrees. How tall is the tree?
Question 3: In the function h(t) = 3cos(2t) + 5, what is the amplitude and what is the vertical shift?
Question 4: If the angle of depression from point A to point B is 28 degrees, what is the angle of elevation from B to A?
Show Answers
- Use Law of Cosines when you have a non-right triangle (specifically when you know two sides and the included angle, or all three sides).
- height = 25 * tan(53 degrees) = 33.2 meters
- Amplitude = 3, Vertical shift = 5
- 28 degrees (angles of depression and elevation are equal due to alternate interior angles)
Next Steps
- Practice drawing clear diagrams for word problems
- Review the Law of Sines and Law of Cosines formulas
- Move on to Common Mistakes to learn what to avoid
- Create your own word problems for additional practice