Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is widely used in various fields like engineering, physics, architecture, and navigation. One of the most common applications of trigonometry is in solving word problems. These problems involve real-w...
Trigonometric word problems often involve concepts like:
- Angle of Elevation: The angle formed between the horizontal line and the line of sight when looking upwards.
- Angle of Depression: The angle formed between the horizontal line and the line of sight when looking downwards.
- SOH CAH TOA: A mnemonic used to remember the trigonometric ratios:
- Sine (SOH) = Opposite / Hypotenuse
- Cosine (CAH) = Adjacent / Hypotenuse
- Tangent (TOA) = Opposite / Adjacent
Solving Trigonometric Word Problems: A Step-by-Step Guide
To solve trigonometric word problems effectively, follow these steps:
- Read the problem carefully and identify the known and unknown quantities. This includes understanding the context of the problem and what you are trying to find.
- Draw a diagram representing the situation described in the problem. Label the diagram with the known and unknown quantities, including the angles and sides.
- Choose the appropriate trigonometric ratio based on the known and unknown quantities. Use SOH CAH TOA to determine which trigonometric function (sine, cosine, or tangent) relates the known angle and sides to the unknown quantity.
- Set up an equation using the chosen trigonometric ratio. Substitute the known values into the equation.
- Solve the equation for the unknown quantity. This may involve using inverse trigonometric functions to find the angle or algebraic manipulations to solve for the length.
- Check your answer and make sure it makes sense in the context of the problem. Consider if the answer is reasonable and units are correct.
Example Problems and Solutions
Let's illustrate these steps with some examples:
Problem 1: Tree Shadow
If a tree casts a 60 ft long shadow and the angle of elevation from the end of the shadow to the top of the tree is 39.80 degrees, how high is the tree?
Diagram:
Solution:
- Known: Shadow length (adjacent side) = 60 ft, Angle of elevation = 39.80 degrees.
Unknown: Tree height (opposite side).
- Diagram: Refer to the diagram above.
- Trigonometric ratio: Since we know the adjacent side and want to find the opposite side, we use the tangent function: tan(angle) = opposite / adjacent.
- Equation: tan(39.80 degrees) = opposite / 60 ft.
- Solve: Opposite = tan(39.80 degrees) * 60 ft ≈ 49.6 ft.
- Answer: The tree is approximately 49.6 ft high.
Problem 2: Building and Louisse
Louisse is at a distance X from the base of a 50 ft high building. She notices that looking at the top of the building, the angle of elevation is 41 degrees. How far is she from the top of the building and from the base of the building (X)?
Diagram:
Solution:
- Known: Building height (opposite side) = 50 ft, Angle of elevation = 41 degrees.
Unknown: Distance from Louisse to the base of the building (adjacent side, X), Distance from Louisse to the top of the building (hypotenuse).
- Diagram: Refer to the diagram above.
- Trigonometric ratio: To find X (adjacent side), use the tangent function: tan(angle) = opposite / adjacent.
- Equation: tan(41 degrees) = 50 ft / X.
- Solve: X = 50 ft / tan(41 degrees) ≈ 57.8 ft.
- To find the distance from Louisse to the top of the building: Use the Pythagorean theorem: Hypotenuse^2 = Opposite^2 + Adjacent^2.
Hypotenuse^2 = 50^2 + 57.8^2 ≈ 5848.84.
Hypotenuse ≈ 76.4 ft.
- Answer: Louisse is approximately 57.8 ft from the base of the building and 76.4 ft from the top of the building.
Problem 3: Airplane and Runway
An airplane is flying at an altitude of 2 miles from the ground and 5 miles away from the airport runway. What is the angle of depression of the plane to the runway of the airport?
Diagram:
Solution:
- Known: Altitude (opposite side) = 2 miles, Distance from the runway (adjacent side) = 5 miles.
Unknown: Angle of depression.
- Diagram: Refer to the diagram above.
- Trigonometric ratio: Since we know the opposite and adjacent sides, we use the tangent function: tan(angle) = opposite / adjacent.
- Equation: tan(angle) = 2 miles / 5 miles.
- Solve: Angle = arctan(2/5) ≈ 21.8 degrees.
- Answer: The angle of depression of the plane to the runway is approximately 21.8 degrees.
Problem 4: Office Buildings
Two office buildings are 60 meters apart. The shorter building is 250 meters high. The angle of depression from the top of the taller building to the top of the smaller building is 20 degrees. How high is the taller building?
Diagram:
Solution:
- Known: Distance between buildings (adjacent side) = 60 meters, Height of shorter building (opposite side) = 250 meters, Angle of depression = 20 degrees.
Unknown: Height of taller building.
- Diagram: Refer to the diagram above.
- Trigonometric ratio: To find the height difference between the buildings, use the tangent function: tan(angle) = opposite / adjacent.
- Equation: tan(20 degrees) = height difference / 60 meters.
- Solve: Height difference = tan(20 degrees) * 60 meters ≈ 21.9 meters.
- To find the height of the taller building: Add the height difference to the height of the shorter building.
Height of taller building = 250 meters + 21.9 meters ≈ 271.9 meters.
- Answer: The taller building is approximately 271.9 meters high.
Conclusion
Solving trigonometric word problems involves understanding the concepts of angles of elevation and depression, applying SOH CAH TOA, and using trigonometric functions to relate known and unknown quantities in triangles. By following a systematic approach of drawing diagrams, choosing the correct trigonometric ratio, setting up equations, and solving for the unknowns, you can effectively solve various real-world problems involving triangles.