Examples of Trigonometric Ratio Word Problems
Trigonometric ratios relate to the angles and sides of right triangles. The primary trigonometric ratios are:
- Sine (sin): sin(θ) = opposite/hypotenuse
- Cosine (cos): cos(θ) = adjacent/hypotenuse
- Tangent (tan): tan(θ) = opposite/adjacent
Let's go through a few examples to better understand how these ratios are applied in word problems.
Example 1: Finding Height
A 14-foot ladder is placed against a wall, forming a 70° angle with the ground. How high does the ladder reach on the wall?
To solve this, we identify:
- Opposite side (height of the wall) = ?
- Hypotenuse (length of the ladder) = 14 ft
- Angle θ = 70°
We can use the sine ratio:
sin(θ) = opposite/hypotenuse
sin(70°) = height/14
height = 14 * sin(70°)
Now calculate:
height ≈ 14 * 0.9397 ≈ 13.16 ft
The ladder reaches approximately 13.16 feet up the wall.
Example 2: Finding Distance
A surveyor is standing 100 meters from the base of a hill. She measures the angle of elevation to the top of the hill as 30°. How tall is the hill?
In this case:
- Opposite side (height of the hill) = ?
- Adjacent side (distance from the hill) = 100 m
- Angle θ = 30°
We use the tangent ratio:
tan(θ) = opposite/adjacent
tan(30°) = height/100
height = 100 * tan(30°)
Now calculate:
height ≈ 100 * 0.5774 ≈ 57.74 m
The hill is approximately 57.74 meters tall.
Example 3: Finding Angle
A person stands 50 meters away from a tree and looks up to the top of the tree at an angle of elevation of 45°. What is the height of the tree?
Here, we know:
- Opposite side (height of the tree) = ?
- Adjacent side (distance from the tree) = 50 m
- Angle θ = 45°
We use the tangent ratio again:
tan(θ) = opposite/adjacent
tan(45°) = height/50
height = 50 * tan(45°)height = 50 * 1 = 50 m
The height of the tree is 50 meters.
Conclusion
These examples illustrate how trigonometric ratios can be used to solve real-life problems involving right triangles. By identifying the sides and the angle, we can utilize sine, cosine, and tangent to find missing measurements.