Online Tangent Calculator | Calculate tan(x) and arctan(x)

What is the Tangent Function?

Definition in a right-angled triangle
The tangent function on the unit circle
Key properties like period, domain, and range

The tangent function, denoted as tan(x), is one of the primary trigonometric functions. In the context of a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. This can be remembered by the mnemonic SOH-CAH-TOA.

tan(θ) = Opposite / Adjacent

The tangent can also be defined using the unit circle as the ratio of the sine to the cosine of an angle (tan(x) = sin(x) / cos(x)). This definition extends the function to all real numbers. The value of the tangent represents the slope of the line segment from the origin to the point on the unit circle corresponding to the angle.

The tangent function has a period of π (or 180°), meaning its values repeat every π radians. It is undefined at angles where the cosine is zero, such as ±π/2, ±3π/2, etc. (or ±90°, ±270°, etc.). The range of the tangent function is all real numbers.

Fundamental Concepts

tan(45°) = 1
tan(0) = 0
tan(60°) = √3 ≈ 1.732
tan(π/4) = 1

Step-by-Step Guide to Using the Tangent Calculator

Calculating the tangent of an angle
Calculating the inverse tangent (arctangent)
Switching between degrees and radians

This calculator is designed to be simple and intuitive. Follow these steps to get your result:

1. Select the Calculation Type

Choose 'Find Tangent of an Angle' if you have an angle and want to find its tangent. Choose 'Find Arctangent of a Value' if you have a value and want to find the corresponding angle (this is the inverse operation).

2. Enter Your Input

If finding the tangent, enter the angle in the 'Angle (x)' field and select whether the unit is 'Degrees' or 'Radians'.

If finding the arctangent, enter the numeric value in the 'Value' field.

3. Calculate and Interpret the Result

Click the 'Calculate' button. The result will be displayed below. For arctangent, the result is given in both degrees and radians. Use the 'Reset' button to clear all inputs.

Practical Usage

To find tan(30°), select 'Find Tangent', enter 30, select 'Degrees', and calculate.
To find the angle whose tangent is 2, select 'Find Arctangent', enter 2, and calculate.

Real-World Applications of the Tangent Function

Architecture and Engineering
Navigation and Astronomy
Physics and Computer Graphics

The tangent function is not just an abstract concept; it has numerous practical applications.

Architecture and Engineering

Architects and engineers use the tangent function to calculate angles of inclination, slopes of roofs, and forces acting on structures. For example, calculating the slope of a ramp or a hill involves the tangent.

Navigation and Astronomy

In navigation, the tangent is used to determine the distance to an object of known height by measuring the angle of elevation. Astronomers use it to calculate the distance or size of celestial bodies.

Physics and Computer Graphics

In physics, the tangent helps describe the trajectory of projectiles and wave phenomena. In computer graphics, it's essential for rendering 3D objects, calculating light reflection angles, and rotating objects.

Application Examples

Calculating the height of a building by measuring the angle of elevation from a certain distance.
Determining the pitch of a roof in construction.
In video games, to determine an object's orientation relative to the camera.

Understanding Arctangent (Inverse Tangent)

What arctangent represents
The principal value range
Difference between atan and atan2

The inverse tangent, or arctangent (denoted as arctan(y), atan(y), or tan⁻¹(y)), is the inverse function of the tangent. It answers the question: 'Which angle has a tangent of y?'

Principal Value

Since the tangent function is periodic, there are infinitely many angles that have the same tangent value. To make the arctangent a well-defined function, we restrict its output to a specific range, known as the principal value. The standard range for arctan(y) is from -π/2 to π/2 radians (or -90° to 90°).

Arctan vs. Atan2

While this calculator computes atan(y), another related function, atan2(y, x), is common in programming. Atan2 takes two arguments (the y and x coordinates) and returns an angle between -π and π, correctly identifying the quadrant of the angle. This is useful when converting from Cartesian to polar coordinates.

Arctangent Examples

arctan(1) = 45° or π/4 radians.
arctan(0) = 0° or 0 radians.
arctan(-√3) = -60° or -π/3 radians.

Common Misconceptions and Important Notes

tan⁻¹(x) vs. 1/tan(x)
The role of radians in calculus
Handling undefined values

Inverse vs. Reciprocal

A common point of confusion is the notation tan⁻¹(x). This represents the inverse tangent (arctan), not the multiplicative reciprocal 1/tan(x). The reciprocal of the tangent function is the cotangent function, cot(x).

Degrees vs. Radians

While degrees are common in everyday life, radians are the standard unit for angles in higher mathematics, particularly calculus and physics. This is because formulas for derivatives and integrals of trigonometric functions are much simpler when expressed in radians.

Undefined Points

Remember that tan(x) is undefined for x = 90° + 180°k (where k is an integer), because cos(x) is zero at these angles, leading to division by zero. Our calculator will show an error for these inputs.

Key Takeaways

tan⁻¹(x) ≠ 1/tan(x)
Always check your unit (degrees/radians) before calculation.
The tangent function has vertical asymptotes where the cosine is zero.

Tangent of 45°

Tangent of π/4 Radians

Arctangent of 1

Arctangent of -0.5