Tangent

Definition

A tangent is a straight 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 and geometry, with applications across various mathematical disciplines.

5 Must Know Facts For Your Next Test

1. The tangent function, denoted as $\tan(x)$, is one of the three primary trigonometric functions, along with sine and cosine.
2. In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.
3. The tangent function is periodic, with a period of $\pi$ (180 degrees).
4. The tangent function is undefined when the angle is an odd multiple of $\frac{\pi}{2}$ (90 degrees), as the adjacent side becomes zero.
5. The tangent function is used extensively in solving right triangle problems, as well as in analyzing the behavior of periodic functions.

Review Questions

• Explain how the tangent function is defined and calculated in the context of a right triangle.
  • In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. Specifically, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. This means that to find the tangent of an angle, you need to identify the opposite and adjacent sides of the right triangle and divide the opposite side by the adjacent side. This relationship is fundamental to solving problems involving right triangles and using trigonometric functions.
• Describe how the tangent function is represented and used in the unit circle.
  • The unit circle is a crucial tool for understanding and visualizing trigonometric functions, including the tangent function. On the unit circle, the tangent function is represented by the slope of the line tangent to the circle at a given angle. The tangent function is defined as the ratio of the $y$-coordinate to the $x$-coordinate of a point on the unit circle. This allows us to use the unit circle to determine the values of the tangent function for different angles and to analyze its periodic behavior.
• Explain how the tangent function is used in verifying trigonometric identities and simplifying trigonometric expressions.
  • The tangent function is an essential component in verifying trigonometric identities and simplifying trigonometric expressions. Many trigonometric identities, such as the Pythagorean identities and the sum and difference formulas, involve the tangent function. By understanding the properties and relationships of the tangent function, you can manipulate and simplify complex trigonometric expressions, which is a crucial skill in solving a variety of mathematical problems.

Related terms

Trigonometric Functions: The trigonometric functions, such as sine, cosine, and tangent, are mathematical functions that describe the relationships between the sides and angles of a right triangle.

Unit Circle: The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane, and is used to define and visualize trigonometric functions.

Inverse Trigonometric Functions: The inverse trigonometric functions, such as arctan or tan^(-1), are used to find the angle given the value of a trigonometric function.