The Tangent Inverse Calculator (also known as Arc Tangent or Arctan) is used to find the measure of an angle in a right-angled triangle when the lengths of the opposite side and the adjacent side are known. Simply enter the side lengths to get the angle in both degrees and radians.
tan inverse calculator
tan inverse calculator Formula
The core formula for finding the angle $\theta$ using the tangent inverse function is:
$$\theta = \arctan\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)$$
Where $\arctan$ is the inverse tangent function, and:
- Opposite (Y) is the length of the side opposite the angle $\theta$.
- Adjacent (X) is the length of the side adjacent to the angle $\theta$.
Formula Source: Khan Academy – Inverse Trigonometric Functions | Wolfram MathWorld
Variables Used in the Calculator
- Opposite Side (Y): The length of the side opposite the unknown angle $\theta$ (typically the vertical coordinate). Must be a non-negative number.
- Adjacent Side (X): The length of the side adjacent to the unknown angle $\theta$ (typically the horizontal coordinate). Must be a positive number.
What is tan inverse calculator?
The Tangent Inverse Calculator, often written as $\tan^{-1}(x)$ or $\arctan(x)$, is a fundamental tool in trigonometry used to reverse the tangent operation. While the tangent function takes an angle and returns the ratio of the opposite to the adjacent side, the inverse tangent function takes that ratio and returns the corresponding angle.
This is crucial in geometry, physics, and engineering when you need to find direction, slope, or the angle of elevation/depression from known distances. In coordinate geometry, it’s used to determine the angle a vector makes with the positive x-axis.
How to Calculate tan inverse calculator (Example)
- Identify the Sides: Suppose you have a right triangle where the Opposite Side (Y) is $15$ and the Adjacent Side (X) is $10$.
- Set up the Ratio: Calculate the tangent ratio: $$\text{Ratio} = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{15}{10} = 1.5$$
- Apply the Inverse Function: Use the arctan function on the ratio: $$\theta = \arctan(1.5)$$
- Calculate the Angle: The calculator performs this step. The result in radians is approximately $0.9828$ radians.
- Convert to Degrees: Convert the radian value to degrees (by multiplying by $\frac{180}{\pi}$): $$\theta \approx 0.9828 \times \frac{180}{\pi} \approx 56.31^\circ$$
- Result: The angle $\theta$ is $56.31$ degrees.
Frequently Asked Questions (FAQ)
- What is the difference between $\tan$ and $\tan^{-1}$?
The tangent ($\tan$) function takes an angle and returns a ratio. The inverse tangent ($\tan^{-1}$ or $\arctan$) function takes a ratio and returns the angle that produced that ratio. They are inverse operations. - What units does $\tan^{-1}$ typically return?
Most scientific calculators and programming languages return the result in radians. This calculator provides the result in both radians and the more commonly used degrees. - Can the Adjacent Side (X) be zero?
No. If the Adjacent Side (X) is zero, the ratio is undefined, leading to an infinite slope or a $90^\circ$ angle. Our calculator will produce an error message for $X=0$ to prevent division by zero. - What is the range of the $\arctan$ function?
For the principal value (which is what standard calculators return), the angle $\theta$ is in the range of $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ (or $-90^\circ$ to $90^\circ$).
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