Tangent

Tangent to a curve

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Tangent circles

Tangent refers to a concept in geometry that has applications in various branches of mathematics, including trigonometry and calculus. The term originates from the Latin word tangens, which means "touching", aptly describing how a tangent line touches a curve at a single point. This article will explore the definition of tangent in different mathematical contexts, its properties, and applications.

Definition

In geometry, a tangent to a circle is a straight line that touches the circle at exactly one point. This point is known as the point of tangency. The concept can be extended to other curves, where a tangent line at a given point on the curve is a straight line that just "touches" the curve at that point. This line is perpendicular to the radius at the point of tangency in the case of a circle.

In trigonometry, the tangent of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. This definition is often abbreviated as "tan", such as in the expression tan(θ) = opposite/adjacent, where θ is the angle.

In calculus, the tangent line to a curve at a given point is the best linear approximation of the curve near that point. The slope of this tangent line is equal to the derivative of the curve's equation at that point.

Properties

• In a circle, a tangent is perpendicular to the radius at the point of tangency.
• The tangent function in trigonometry is periodic, with a period of π radians (180 degrees), meaning that it repeats its values every π radians.
• In calculus, the slope of the tangent line to a curve at a given point provides instantaneous rate of change of the curve at that point.

Applications

Tangents have numerous applications in both pure and applied mathematics, as well as in fields such as engineering, physics, and architecture. Some applications include:

• Determining the slope of a curve at a given point, which is essential in the study of motion and rates of change in physics.
• Designing curved paths and surfaces in architecture and engineering where a certain tangent slope is required at a point of contact.
• Calculating angles and distances in trigonometry that are used in navigation, surveying, and astronomy.

See Also

• Circle
• Trigonometry
• Calculus
• Derivative
• Slope