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Calculating Cotangent of an Angle
The cotangent (cot) of an angle in a right-angled triangle is a trigonometric function that represents the ratio of the length of the adjacent side to the opposite side. The cotangent function is the reciprocal of the tangent function and is often used in trigonometric calculations.
The cotangent of an angle \( \theta \) is defined as:
\( \cot(\theta) = \dfrac{\text{length of adjacent side}}{\text{length of opposite side}} \)
where:
- adjacent side is the side next to the angle \( \theta \), excluding the hypotenuse.
- opposite side is the side opposite to angle \( \theta \).
Consider the following triangle ABC, right-angled at vertex B.
For the angle \( \theta \) at vertex A:
- Adjacent side = AB
- Opposite side = BC
Therefore,
\( \cot(\theta) = \dfrac{\text{length of side AB}}{\text{length of side BC}} \)
Examples
The following examples demonstrate how to use the cotangent of an angle to find the length of the adjacent side or the opposite side in various practical situations.
1. What is the cotangent of an angle θ if the length of the adjacent side is 6 units and the length of the opposite side is 2 units?
Answer
Given:
- Length of the adjacent side = 6 units
- Length of the opposite side = 2 units
The formula to calculate the cotangent of an angle θ is:
\( \cot(\theta) = \dfrac{\text{length of adjacent side}}{\text{length of opposite side}} \)
Substitute the given values into the formula:
\( \cot(\theta) = \dfrac{6}{2} \)
Simplify the expression:
\( \cot(\theta) = 3 \)
∴ cot(θ) = 3
2. A ramp rises at an angle of 45° to the horizontal. If the vertical height of the ramp is 10 meters, what is the horizontal distance covered by the ramp?
Answer
Given:
- Vertical height (opposite side) = 10 meters
- Angle with the horizontal θ = 45°
Using the cotangent function to find the horizontal distance (adjacent side):
\( \cot(45°) = \dfrac{\text{horizontal distance}}{10} \)
Simplify the expression:
\( 1 = \dfrac{\text{horizontal distance}}{10} \)
Now solve for the horizontal distance:
\( \text{Horizontal distance} = 1 \times 10 = 10 \text{ meters} \)
∴ The ramp covers a horizontal distance of 10 meters.
3. A tree casts a shadow that is 30 meters long. If the angle of elevation of the sun is 30°, how tall is the tree?
Answer
Given:
- Length of the shadow (adjacent side) = 30 meters
- Angle of elevation θ = 30°
Using the cotangent function to find the height of the tree (opposite side):
\( \cot(30°) = \dfrac{30}{\text{height}} \)
Simplify the expression:
\( 1.732 = \dfrac{30}{\text{height}} \)
Now solve for the height:
\( \text{Height} = \dfrac{30}{1.732} \approx 17.32 \text{ meters} \)
∴ The tree is approximately 17.32 meters tall.
4. A ladder is leaning against a building at an angle of 60° to the ground. If the distance from the base of the ladder to the wall is 5 meters, what is the height at which the ladder touches the wall?
Answer
Given:
- Distance from the wall (adjacent side) = 5 meters
- Angle with the ground θ = 60°
Using the cotangent function to find the height where the ladder touches the wall (opposite side):
\( \cot(60°) = \dfrac{5}{\text{height}} \)
Simplify the expression:
\( 0.577 = \dfrac{5}{\text{height}} \)
Now solve for the height:
\( \text{Height} = \dfrac{5}{0.577} \approx 8.66 \text{ meters} \)
∴ The height at which the ladder touches the wall is approximately 8.66 meters.