Arctangent

Definition

The arctangent, also known as the inverse tangent, is a trigonometric function that calculates the angle whose tangent is a given value. It is used to determine the angle in a right-angled triangle when the length of the opposite and adjacent sides are known.

5 Must Know Facts For Your Next Test

1. The arctangent function is denoted as $\arctan$ or $\tan^{-1}$.
2. The arctangent function is used to find the angle in a right-angled triangle when the lengths of the opposite and adjacent sides are known.
3. The arctangent function returns an angle between $-\pi/2$ and $\pi/2$ radians, or between $-90$ and $90$ degrees.
4. The arctangent function is the inverse of the tangent function, meaning that $\arctan(\tan(x)) = x$.
5. The arctangent function is a monotonically increasing function, meaning that as the input value increases, the output angle also increases.

Review Questions

• Explain how the arctangent function is used to determine the angle in a right-angled triangle.
  • In a right-angled triangle, the arctangent function can be used to find the angle when the lengths of the opposite and adjacent sides are known. The formula for the arctangent is $\theta = \arctan(\text{opposite}/\text{adjacent})$, where $\theta$ is the angle in the triangle. By plugging in the known side lengths, the arctangent function can be used to calculate the angle of the triangle.
• Describe the properties of the arctangent function and how they relate to its use in free-body diagrams.
  • The arctangent function has several important properties that make it useful in the context of free-body diagrams. First, it returns an angle between $-\pi/2$ and $\pi/2$ radians, which corresponds to the range of possible angles in a free-body diagram. Additionally, the arctangent function is monotonically increasing, meaning that as the input value (the ratio of the opposite and adjacent sides) increases, the output angle also increases. This allows for a direct mapping between the side lengths and the angle in a free-body diagram.
• Analyze how the use of the arctangent function in free-body diagrams can help in the analysis of forces and their directions.
  • In free-body diagrams, the arctangent function is crucial for determining the direction of forces acting on an object. By calculating the arctangent of the ratio of the opposite and adjacent sides, the angle of a force can be determined. This information is essential for resolving the force into its horizontal and vertical components, which is a key step in analyzing the net forces acting on an object and determining its acceleration or equilibrium conditions. The arctangent function provides a direct link between the geometry of the free-body diagram and the trigonometric relationships needed to analyze the forces.

Related terms

Tangent:

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side.

Inverse Trigonometric Functions: Inverse trigonometric functions, such as the arctangent, are used to find the angle given the value of a trigonometric ratio.

Right-Angled Triangle: A right-angled triangle is a triangle in which one of the angles is a 90-degree angle, or a right angle.