Tangent Calculator

The tangent function, often denoted as tan(x), is a fundamental trigonometric function that relates the ratio of the sine and cosine functions, providing a measure of the slope or steepness of a line in relation to the x-axis.
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tan⁡(1)=  1.557407725\tan({1})=\;1.5574077251.557407725
Photo of calculator author Wes NolteCreated by Wes Nolte.Last updated Apr 25, 2024.

The Tangent Calculator Formula

y=tan⁡(x)y = \tan({x})

Definition of the Tangent Function

The tangent function, denoted as tan(x), is a trigonometric function that relates the angle x to the ratio of the sine and cosine of that angle. It is defined as the opposite side divided by the adjacent side of a right triangle. In other words, tan(x) = sin(x) / cos(x).The tangent function is periodic with a period of π radians or 180 degrees, which means its values repeat after every half-rotation around the unit circle. It has vertical asymptotes at odd multiples of π/2, where the cosine function becomes zero, resulting in an undefined value for the tangent.The tangent function exhibits several key properties. It is an odd function, meaning that tan(-x) = -tan(x), and it is also periodic with a phase shift of π radians or 180 degrees. As x approaches these vertical asymptotes, the tangent function tends towards positive or negative infinity, depending on the direction of approach.The values of the tangent function range from negative infinity to positive infinity, as it covers all possible real numbers. The tangent function also has periodicity with π radians or 180 degrees, meaning that tan(x) = tan(x + nπ) for any integer n.The tangent function finds applications in various fields, such as physics, engineering, and mathematics. It is particularly useful in trigonometric calculations involving angles, slopes, and rotations. The tangent function is also utilized in signal processing, where it helps analyze periodic waveforms and their frequency components.Trigonometric identities involving the tangent function, such as the Pythagorean identity (1 + tan^2(x) = sec^2(x)), and the angle addition and subtraction formulas, contribute to solving equations and simplifying trigonometric expressions.Trigonometric identities involving the tangent function, such as the Pythagorean identity (1 + tan^2(x) = sec^2(x)), and the angle addition and subtraction formulas, contribute to solving equations and simplifying trigonometric expressions.

History of The Tangent Function

The history of the tangent function can be traced back to ancient civilizations, where the study of angles and triangles played a crucial role in various practical applications. However, it was in ancient Greece that the concept of the tangent function began to emerge.The Greek mathematician Hipparchus (circa 190-120 BCE) is often credited as one of the first to develop tables of chords, which were the predecessors to modern trigonometric tables. These tables provided the lengths of the chords for various angles in a circle. The tangent function can be thought of as the ratio of the length of the chord opposite an angle to the length of the adjacent side of a right triangle.The term "tangent" itself comes from the Latin word "tangens," meaning "touching." It reflects the geometric interpretation of the tangent function as the line that touches or is tangent to a circle at a given point.During the Islamic Golden Age (8th to 14th centuries), Muslim mathematicians made significant advancements in trigonometry, including the tangent function. Scholars such as Abu al-Wafa' al-Buzjani and Nasir al-Din al-Tusi expanded upon the works of ancient Greek mathematicians, introducing new trigonometric concepts and refining trigonometric functions.In Europe, the Renaissance period witnessed further development in trigonometry. Mathematicians such as Nicolaus Copernicus, Johannes Kepler, and Isaac Newton made significant contributions to trigonometric functions, including the tangent. Their work in astronomy and physics relied heavily on trigonometry, which contributed to the understanding and refinement of the tangent function.Throughout history, the tangent function has been essential for various scientific and practical applications. It is used in fields such as physics, engineering, surveying, navigation, and computer graphics. Trigonometric tables and the development of calculators and computers have made the evaluation of tangent values more accessible and efficient.The historical development of the tangent function reflects the ongoing efforts of mathematicians and scientists to understand the relationships between angles, lines, and geometric shapes. The concept of the tangent function has evolved and been refined over time, contributing to the foundation of trigonometry as a vital branch of mathematics.

Relationship to Other Trig Functions

The tangent function (tan) has several important relationships with other trigonometric functions. Here are the key relationships:
  1. Sine Function (sin): The tangent function is defined as the ratio of the sine function to the cosine function. Mathematically, tan(x) = sin(x) / cos(x). This relationship shows that the tangent is dependent on both the sine and cosine functions.
  2. Cosine Function (cos): The tangent function can also be expressed in terms of the cosine function. Mathematically, tan(x) = sin(x) / cos(x) = 1 / cos(x) × sin(x) = (1 / cos(x)) × (sin(x) / 1) = sec(x) / 1 = sec(x), where sec(x) represents the secant function.
  3. Cosecant Function (csc): The cosecant function is the reciprocal of the sine function. As the tangent function is defined as the ratio of the sine to the cosine, it can also be expressed in terms of the cosecant function. Mathematically, tan(x) = sin(x) / cos(x) = (1 / sin(x)) × (sin(x) / cos(x)) = (1 / sin(x)) × csc(x) = csc(x) / sin(x).
  4. Secant Function (sec): The secant function is the reciprocal of the cosine function. Since the tangent function can be expressed as the ratio of the sine to the cosine, it can also be written in terms of the secant function. Mathematically, tan(x) = sin(x) / cos(x) = sin(x) × (1 / cos(x)) = sin(x) × sec(x) = sec(x) × sin(x).
  5. Cotangent Function (cot): The cotangent function is the reciprocal of the tangent function. Mathematically, cot(x) = 1 / tan(x). This relationship shows that the cotangent can be expressed as the inverse of the tangent.