Tan
Calculate the tangent (tan) of any angle in degrees or radians instantly. Free online tangent calculator for students, engineers, and trigonometry professionals.
About This Calculator
The tangent (tan) calculator computes the tangent of any angle in degrees or radians. Tangent is one of the six fundamental trigonometric functions and is defined as the ratio of the opposite side to the adjacent side in a right triangle: tan(theta) = opposite / adjacent = sin(theta) / cos(theta). This calculator is designed for students learning trigonometry, engineers solving angle problems, architects calculating slopes, and professionals working with angles in various fields.
How the Tangent is Calculated
The tangent of an angle theta in a right triangle is the length of the side opposite theta divided by the length of the side adjacent to theta. The tangent function is periodic with a period of 180 deg (pi radians). It crosses zero at every multiple of 180 deg (0 deg, 180 deg, 360 deg, etc.) and has vertical asymptotes at 90 deg, 270 deg, and so on, where the function approaches +/-∞.
Mathematically, the tangent is expressed as tan(theta) = sin(theta) / cos(theta). This identity is fundamental in trigonometry and can be derived from the unit circle definition where sin(theta) is the y-coordinate and cos(theta) is the x-coordinate of a point on the unit circle. The tangent then represents the slope of the line from the origin to that point.
How to Use This Calculator
Enter any angle in the input field and select degrees or radians from the dropdown menu. Click Calculate to compute the tangent value. The result displays the tangent of the input angle. You can toggle between degrees and radians at any time. Common angles to try: 45 deg (tan = 1), 30 deg (tan ≈ 0.577), 60 deg (tan ≈ 1.732), 0 deg (tan = 0), and 90 deg (tan is undefined/approaches infinity).
Frequently Asked Questions
What is the tangent of an angle?
In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. It is one of the six fundamental trigonometric functions and is defined as tan(theta) = sin(theta) / cos(theta). The tangent function repeats every 180 deg (pi radians) and has vertical asymptotes where cos(theta) = 0.
What is tan 45 degrees?
tan(45 deg) = 1. This is because at 45 degrees in a right triangle, the opposite and adjacent sides are equal in length, so their ratio is 1. This is one of the most commonly used values in trigonometry.
When is tangent undefined?
Tangent is undefined when the cosine of the angle equals zero, which occurs at 90 deg (pi/2 radians), 270 deg (3pi/2 radians), and every 180 deg from those points. At these angles, the tangent function has vertical asymptotes and the value approaches positive or negative infinity.
What is the difference between tan in degrees and radians?
Degrees and radians are two different units for measuring angles. 180 deg equals pi radians. The tangent of an angle depends on the unit used. For example, tan(45 deg) = 1, while tan(45 radians) ≈ 1.62. Always ensure you are using the correct unit for your calculation. Use the dropdown to switch between deg and rad.
What is the range of the tangent function?
The range of the tangent function is all real numbers (-∞ to ∞). Unlike sine and cosine which are bounded between -1 and 1, the tangent function can take any real value. It increases from -∞ to +∞ between each pair of vertical asymptotes.
How is tangent used in real life?
Tangent is widely used in engineering, physics, architecture, and navigation. Common applications include calculating slopes and gradients (road grades, roof pitches), determining heights of buildings or mountains using angle measurements (clinometer), analyzing wave functions in signal processing, and computing angles in computer graphics and game development.
Is this tool free to use?
Yes, all calculators on Calculy are completely free to use. No registration, login, or payment is required. You can also share your calculations with others using the share button.