ConvertOnlinetan−1 Calculator
x= opposite_side/adjacent_side
tan−1(x)
Calculating Inverse Tangent of a Value
The inverse tangent (tan⁻¹) function, also known as arctangent, is used to find the angle whose tangent is a given value. This function is particularly useful in trigonometry when you need to determine the angle based on the ratio of the opposite side to the adjacent side in a right-angled triangle.
The inverse tangent of a value \( x \) is defined as:
\( \theta = \tan^{-1}(x) \)
where:
- θ is the angle corresponding to the tangent value.
- x is the tangent value, representing the ratio of the opposite side to the adjacent side.
Consider the following triangle ABC, right-angled at vertex B.
For the angle \( \theta \) at vertex A:
- Opposite side = BC
- Adjacent side = AB
Therefore,
\( \tan^{-1}(x) = \theta \), where \( x = \dfrac{\text{length of side BC}}{\text{length of side AB}} \)
Examples
The following examples demonstrate how to use the inverse tangent function to find the angle when the tangent value is known.
1. A streetlight casts a shadow 15 feet long on the ground. If the height of the streetlight is 12 feet, what is the angle of elevation of the light from the tip of the shadow?
Answer
Given:
- Length of the opposite side (height of the streetlight) = 12 feet
- Length of the adjacent side (length of the shadow) = 15 feet
First, calculate the tangent of the angle:
\( \tan(\theta) = \dfrac{\text{height of streetlight}}{\text{length of shadow}} = \dfrac{12}{15} \)
Simplify the expression:
\( \tan(\theta) = 0.8 \)
Now, find the angle using the inverse tangent function:
\( \theta = \tan^{-1}(0.8) \)
Using a calculator or reference table, the angle is:
∴ θ ≈ 38.7°
2. A firefighter is using a ladder to reach a window on the second floor of a building. The ladder is placed 10 meters away from the base of the building, and the window is 15 meters above the ground. What is the angle at which the ladder is inclined?
Answer
Given:
- Length of the opposite side (height to the window) = 15 meters
- Length of the adjacent side (distance from the building) = 10 meters
First, calculate the tangent of the angle:
\( \tan(\theta) = \dfrac{\text{height to the window}}{\text{distance from the building}} = \dfrac{15}{10} \)
Simplify the expression:
\( \tan(\theta) = 1.5 \)
Now, find the angle using the inverse tangent function:
\( \theta = \tan^{-1}(1.5) \)
Using a calculator or reference table, the angle is:
∴ θ ≈ 56.3°
3. A camera is set up to capture the top of a tall monument. If the camera is positioned 20 meters away from the base of the monument and the height of the monument is 50 meters, what is the angle of elevation of the camera?
Answer
Given:
- Length of the opposite side (height of the monument) = 50 meters
- Length of the adjacent side (distance from the monument) = 20 meters
First, calculate the tangent of the angle:
\( \tan(\theta) = \dfrac{\text{height of monument}}{\text{distance from monument}} = \dfrac{50}{20} \)
Simplify the expression:
\( \tan(\theta) = 2.5 \)
Now, find the angle using the inverse tangent function:
\( \theta = \tan^{-1}(2.5) \)
Using a calculator or reference table, the angle is:
∴ θ ≈ 68.2°
4. A car is driving up a hill that rises 8 meters vertically for every 30 meters of horizontal distance. What is the angle of the hill's incline?
Answer
Given:
- Length of the opposite side (vertical rise) = 8 meters
- Length of the adjacent side (horizontal distance) = 30 meters
First, calculate the tangent of the angle:
\( \tan(\theta) = \dfrac{\text{vertical rise}}{\text{horizontal distance}} = \dfrac{8}{30} \)
Simplify the expression:
\( \tan(\theta) = 0.267 \)
Now, find the angle using the inverse tangent function:
\( \theta = \tan^{-1}(0.267) \)
Using a calculator or reference table, the angle is:
∴ θ ≈ 15.0°