Tangent

Definition

A tangent is a line that touches a curve at a single point, forming a right angle with the curve at that point. It is a fundamental concept in trigonometry, geometry, and calculus, with applications across various mathematical and scientific disciplines.

5 Must Know Facts For Your Next Test

1. The tangent function is one of the six fundamental trigonometric functions, along with sine, cosine, cotangent, secant, and cosecant.
2. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
3. The tangent function is used to solve problems involving right triangles, such as finding the unknown side lengths or angle measures.
4. The graph of the tangent function is a periodic function, repeating every $180^\circ$ or $\pi$ radians.
5. Tangent lines are important in calculus, as they are used to approximate the behavior of a function near a point and to find the slopes of curves.

Review Questions

• Explain how the tangent function is defined in the context of right triangle trigonometry.
  • In the context of right triangle trigonometry, the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. Specifically, the tangent of an angle $\theta$ is equal to the length of the opposite side divided by the length of the adjacent side: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. This definition allows us to use the tangent function to solve for unknown side lengths or angle measures in right triangles.
• Describe the graphical properties of the tangent function and how they relate to the behavior of the function.
  • The graph of the tangent function is a periodic function, repeating every $180^\circ$ or $\pi$ radians. The function has vertical asymptotes at odd multiples of $90^\circ$ or $\frac{\pi}{2}$ radians, where the angle approaches these values, the tangent function approaches positive or negative infinity. Between these asymptotes, the function oscillates between positive and negative values, reflecting the cyclical nature of the tangent function in right triangle trigonometry. These graphical properties are directly related to the underlying mathematical definition of the tangent function and its behavior in different angle ranges.
• Explain the importance of tangent lines in calculus and how they are used to approximate the behavior of a function near a point.
  • In calculus, tangent lines are crucial for approximating the behavior of a function near a specific point. The slope of the tangent line at a point on a curve is equal to the derivative of the function at that point. This allows us to use the tangent line as a linear approximation of the function, which is particularly useful for analyzing the local behavior of the function, such as rates of change, optimization, and the study of limits. Tangent lines are essential tools in differential calculus, as they provide a way to understand the instantaneous rate of change of a function and its graphical properties at a given point.

Related terms

Angle: The measure of the rotation between two intersecting lines or between two planes. Angles are essential in understanding trigonometric functions, including the tangent function.

Trigonometric Functions: The set of functions, including sine, cosine, tangent, cotangent, secant, and cosecant, that describe the relationships between the sides and angles of a right triangle.

Right Triangle: A triangle in which one of the angles is a right angle (90 degrees). The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle.