The calculator supports the majority of trigonometric functions; for example, we can calculate the tan, sine, and cosine of an angle using the same functions. The trigonometric function tangent, abbreviated as tan, allows you to calculate the tangent of an angle electronically using object-based metrics, such as radians, degrees and gradians.

What is tangent? Tangent definition, tangent formula

In mathematics, the term tangent has two aspects. Firstly, we use it in geometry to indicate when one object only touches another thing at one point, such as when a line only meets a circle at one spot. Second, Leibniz described it as the line connecting two infinitely close points on a curve.

A definition of tangent in 1828 is “a right line that touches a curve but does not cut it when formed.” Inflexion points can not have tangents under this outdated definition. The scientist disproved it, and modern definitions equal Leibniz’s, defining the tangent line as a curve connecting two infinitely close points.

Tangent is one of the “big three” trigonometric functions, together with sine and cosine, in trigonometry. Like those two functions, we can use it to find the length of a side or an angle in a right triangle. But it’s also distinct from the other two. It features a unique graph design and a trig “identity” that is very helpful.

Tangent is the proportion of the side opposite the angle we know or like to discover over the side that is adjacent to that angle. T-A-N are three letters that represent tangent. The side is touching the curve, NOT the hypotenuse, the opposite of the right, is the neighbouring side.

When we graph tangent functions, the y-values range from negative infinity to positive infinity, with vertical asymptotes indicating where the graph has no points.

Graph of a tangent

A tangent of a curve y = f(x) at a point x = c is a straight line that passes through the point (c, f(c)) on the curve and has the slope f'(c), where f’ is the derivative of f.  

In the 1700s, these methods led to the discovery of differential calculus. A large number of people contributed. By handling a curve characterized by a moving point whose motion is the product of multiple simpler motions, Roberval established a general way of sketching tangents. The Belgian and Netherlands scientists discovered algebraic tangent algorithms. John Wallis and Isaac Barrow made further developments, which led to Isaac Newton’s and Gottfried Leibniz’s theories.

As the second point reaches the first, it can be viewed as the limiting position of straight lines going through the supplied point and a neighbouring point of the curve.

A line can be described by the linear equation ax + by + c = 0 in a coordinate system on a plane. This equation is commonly represented as y = mx + b, where m denotes the slope and b denotes the line’s value crosses the y-axis.

Tangent – sin over cos

The tangent function is among the most popular trigonometric functions, alongside sine and cosine. The length of the opposing side (O) divided by the length of the opposing side (L) corresponding point is the tangent of an angle in a right-angled triangle (A). Inside the formula, we represent it as ‘tan’.


“SOH” – Sine is Opposite over Hypotenuse – is a popular acronym.

Law of tangents

The trigonometric rule of tangents relates the deviations of the sum and the difference of the angles opposing two sides of a planar triangle.

If a,b, and c are the sides facing angles A, B, and C, correspondingly, in any planar triangle ABC, then:

The formula is convenient when working with logarithms.

Table of common tangent values

Angles in Degrees30°45°60°90°
Sin  0½√2/2√3/21
Cos  1√3/2√2/21/20
Tan  0√3/31√3Not defined
Cot  Not defined√31√3/30
Sec  12√3/3√22Not defined
Csc  Not defined√31√3/30
This is a table of common tangent values

Related trigonometric functions

Trigonometric functions are actual functions in mathematics that link the angle of a right-angled triangle to the proportions of two side lengths.

We employ them in all geometry-related fields, including navigation, solid mechanics, celestial mechanics, geodesy, and many more. They are one of the most effortless periodic functions, and as a result, we can use them in Fourier analysis to examine repetitive events.

In modern mathematics, we are mostly using the sine, cosine, and tangent trigonometric functions. We do not usually use the cosecant, secant, and cotangent, which are their reciprocals. These six trigonometric functions have an inverse function and a hyperbolic function analogue.

Only acute angles are defined in the first definitions of trigonometric functions, which are connected to right-angle triangles. Geometrical descriptions utilizing the standard unit circle are frequently used to expand the sine and cosine functions to functions whose domain is the entire natural line; the domain of the other functions is the real line with specific discrete points eliminated.

Infinite series or differential equation solutions are known as trigonometric functions in modern definitions. This permits the domains of sine and cosine functions to be extended to the entire complex plane, as well as the fields of certain other trigonometric functions to be extended towards the complex plane with certain isolated points eliminated.

How to calculate the tangent of an angle?

The trigonometric ratio between the neighbouring and opposing sides of a right triangle containing that angle is the tangent of that vantage point.

tangent = length of the leg opposite to the angle/length of the leg adjacent to the angle

Regardless matter the size of the right triangle, the tangent ratio remains constant. As a result, it’s usually easier to think of a right triangle with a hypotenuse of length 1.

The tangent ratio can alternatively be viewed as a function that takes different values based on the angle’s measurement. We can measure angles in degrees and radians, respectively.

The cotangent ratio of an angle, known as “cot,” is the counterpart of the tangent ratio.

As seen below, we can commonly represent an angle in the “standard position” in trigonometry.

The vertex of the angle (B) is based on the x and y axes within that location. Therefore, one side of the curve is always set along the positive x-axis, that is, in the direction of 3 o’clock along the axis (line BC). We can call it the angle’s first side.

The opposite side of the angle is the terminal side.

The starting side is the side that is fixed along the positive x-axis (BC). Imagine a copy of this side being rotated around the origin to generate the second side, known as the terminal side, to make the angle.

The formula for tangent’s angle

The angle is determined in degrees or radians, and the quantity we rotate is termed the measure of the angle. This measure can be expressed in the following way:

mABC = 54°

It is pronounced as “the measure of angle ABC is 54 degrees”.

We can represent an angle with just a single letter if it is not confusing. For example, in the diagram above, we can refer to the angle as ABC or angle C.

We frequently use Greek letters to name angles in trigonometry. As an example, the letter θ (theta).

Tangent calculator – example of use

A calculator is almost vital for doing the grunt work, even though it won’t help you master the core principles of trigonometry.

Find the angle’s sine, cosine, or tangent. Then, enter the angle’s degree value and press the “sin,” “cos,” or “tan” buttons.

Convert the sine of an angle to the angle’s measurement. Enter the sine value, press the “arcsin” or “sin-1” button.

Convert the angle’s cosine or tangent into the angle’s measure. Enter the value of the cosine or tangent and press the “arccos” or “cos-1” button.

Find out what multiplicative inverses are and how to use them—reversing the numerator and denominator yields the multiplicative inverse of a number. 3 has a multiplicative inverse of 1/3, for instance.

Find out how to use multiplicative inverses in trigonometry. We can divide six trigonometric functions (sine, cosine, tangent, secant, cosecant, and cotangent) into three pairs of inversions. For example, the inverse of cosecant is sine, the inverse of secant is cosine, and the inverse of cotangent is tangent.

To determine the inverse of a sine, cosine, or tangent value, press the 1/x button. If you know the sine of an angle, and it is 0.66803, for example, hit 1/x to retrieve the cosecant of that number.