Arctangent

Definition

The arctangent is the inverse trigonometric function that gives the angle whose tangent is a given value. It is used to find the angle in a right triangle given the ratio of the opposite and adjacent sides.

5 Must Know Facts For Your Next Test

1. The arctangent function is denoted as $\tan^{-1}(x)$ or $\arctan(x)$, and it gives the angle whose tangent is $x$.
2. The arctangent function is used to find the angle in a right triangle when the ratio of the opposite and adjacent sides is known.
3. The domain of the arctangent function is all real numbers, and the range is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
4. The arctangent function is useful in many applications, such as determining the angle of elevation or depression, finding the direction of a vector, and solving for unknown angles in trigonometric equations.
5. The arctangent function is one of the four basic inverse trigonometric functions, along with $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\cot^{-1}(x)$.

Review Questions

• Explain how the arctangent function is used in right triangle trigonometry.
  • In right triangle trigonometry, the arctangent function is used to find the angle of a triangle when the ratio of the opposite and adjacent sides is known. For example, if you know the length of the opposite and adjacent sides of a right triangle, you can use the arctangent function to calculate the angle between those sides. This is useful in many practical applications, such as determining the angle of elevation or depression, or solving for unknown angles in trigonometric equations.
• Describe the properties of the arctangent function and how they relate to inverse trigonometric functions.
  • The arctangent function is one of the four basic inverse trigonometric functions, along with $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\cot^{-1}(x)$. The arctangent function has a domain of all real numbers and a range of $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. This means that for any real number input, the arctangent function will return the angle whose tangent is that value, within the specified range. The properties of the arctangent function, such as its domain and range, are important to understand when working with inverse trigonometric functions and solving trigonometric equations.
• Analyze the role of the arctangent function in practical applications and how it relates to the concepts of right triangle trigonometry and inverse trigonometric functions.
  • The arctangent function is widely used in practical applications that involve right triangle trigonometry and the need to find unknown angles. For example, in engineering and physics, the arctangent function is used to determine the direction of a vector based on the ratio of its components. In surveying and navigation, the arctangent function is used to calculate the angle of elevation or depression, which is important for tasks like measuring the height of an object or the slope of a hill. Additionally, the arctangent function is crucial in solving trigonometric equations, as it allows you to isolate and find the angle given the ratio of the sides. Understanding the properties and applications of the arctangent function, as well as its relationship to right triangle trigonometry and inverse trigonometric functions, is essential for success in these fields and in solving a variety of mathematical problems.

Related terms

Tangent: The trigonometric function that gives the ratio of the opposite side to the adjacent side of a right triangle.

Inverse Trigonometric Functions: Functions that undo the operations of the basic trigonometric functions, allowing you to find the angle given the ratio of the sides.

Right Triangle Trigonometry: The study of the relationships between the sides and angles of right triangles, which forms the basis for the trigonometric functions.