Tangent Calculator

1. What is the Tangent Calculator?

Definition: This calculator computes the tangent (\( \tan(x) \)) of a given angle \( x \). The tangent function is one of the fundamental trigonometric functions, defined as the ratio of the opposite side to the adjacent side in a right triangle, or equivalently as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). It is widely used in mathematics, physics, engineering, and other fields.

Purpose: The tangent function helps describe relationships in right triangles and is useful in applications like navigation, astronomy, and wave analysis.

2. How Does the Calculator Work?

The calculator computes the tangent using the formula:

\( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

In a right triangle, the tangent of an angle \( x \) is the ratio of the opposite side to the adjacent side:

\( \tan(x) = \frac{\text{opposite}}{\text{adjacent}} \)

Steps:

Enter the angle \( x \) and select its unit (degrees, radians, etc.).
The calculator converts the angle to radians, computes \( \sin(x) \) and \( \cos(x) \), and then calculates \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
If \( \cos(x) = 0 \), the tangent is undefined, and an error message is displayed.
The result is displayed with 5 decimal places.

Unit Conversions (Angles):

Degrees: \( \text{rad} = \text{deg} \times \frac{\pi}{180} \)
Radians: No conversion needed
Gradians: \( \text{rad} = \text{gon} \times \frac{\pi}{200} \)
Turns: \( \text{rad} = \text{tr} \times 2\pi \)
Minutes of Arc: \( \text{rad} = \text{deg2rad}(\text{arcmin} / 60) \)
Seconds of Arc: \( \text{rad} = \text{deg2rad}(\text{arcsec} / 3600) \)
Milliradians: \( \text{rad} = \text{mrad} / 1000 \)
Microradians: \( \text{rad} = \text{urad} / 1000000 \)
π Radians: \( \text{rad} = \text{x π rad} \times \pi \)

3. Properties of the Tangent Function

Domain: The tangent function is defined for all angles except where \( \cos(x) = 0 \), i.e., \( x \neq \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
Range: The tangent function can take any real value: \( \tan(x) \in (-\infty, \infty) \).
Periodicity: The tangent function is periodic with a period of \( \pi \) radians (or 180 degrees).
Odd Function: \( \tan(-x) = -\tan(x) \), making it an odd function.
Key Values:
- \( \tan(0) = 0 \)
- \( \tan\left(\frac{\pi}{4}\right) = 1 \) (or \( 45^\circ \))
- \( \tan\left(\frac{\pi}{2}\right) \) is undefined (or \( 90^\circ \))

4. Using the Calculator

Example 1: Calculate \( \tan(45^\circ) \).

Enter \( x = 45 \), with unit "degrees".
Click "Calculate" to compute:
- \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.70711 \)
- \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.70711 \)
- \( \tan(45^\circ) = \frac{\sin(45^\circ)}{\cos(45^\circ)} = 1 \)
- Displayed as 1.00000 (with 5 decimal places).

Example 2: Calculate \( \tan\left(\frac{\pi}{2}\right) \).

Enter \( x = \frac{\pi}{2} \), with unit "radians".
Click "Calculate" to compute:
- \( \cos\left(\frac{\pi}{2}\right) = 0 \)
- \( \tan\left(\frac{\pi}{2}\right) = \frac{\sin\left(\frac{\pi}{2}\right)}{\cos\left(\frac{\pi}{2}\right)} \) is undefined (error displayed).

5. Frequently Asked Questions (FAQ)

Q: What is the tangent function?
A: The tangent function, denoted \( \tan(x) \), is a fundamental trigonometric function. In a right triangle, it is the ratio of the opposite side to the adjacent side. It can also be defined as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

Q: When is the tangent function undefined?
A: The tangent function is undefined when \( \cos(x) = 0 \), which occurs at angles \( x = \frac{\pi}{2} + n\pi \) radians (or \( 90^\circ + n \cdot 180^\circ \)), where \( n \) is an integer.

Q: What are the different angle units?
A: Angles can be measured in:

Degrees: 360° in a full circle.
Radians: \( 2\pi \) in a full circle.
Gradians: 400 gon in a full circle.
Turns: 1 turn is a full circle.
Minutes of Arc: 60 arcmin per degree.
Seconds of Arc: 3600 arcsec per degree.
Milliradians: 1000 mrad per radian.
Microradians: 1000000 urad per radian.
π Radians: Expressed as a multiple of π.