Problem 29 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\tan (\operatorname{Arctan} 3)\)

Step-by-Step Solution

Answer

3

Step 1: Understand Arctan

The function \( \operatorname{Arctan} \) is the inverse of the tangent function. It returns the angle whose tangent value is the input number. Thus, \( \operatorname{Arctan} 3 \) finds an angle \( \theta \) such that \( \tan(\theta) = 3 \).

Step 2: Simplify the Expression

The given expression is \( \tan (\operatorname{Arctan} 3) \). Essentially, it asks for the tangent of the angle whose tangent is 3, which simplifies directly back to 3. This is because \( \tan(\theta) = 3 \) for the angle found by \( \operatorname{Arctan} 3 \).

Step 3: Solution Verification

Double-check the circular reasoning: since \( \operatorname{Arctan} \) was used to denote an angle whose tangent is 3, applying \( \tan \) to it gives back the original input of 3. Thus, \( \tan (\operatorname{Arctan} 3) = 3 \).

Key Concepts

Arctan

The \text{Arctan}, or inverse tangent, function is a crucial concept in trigonometry. It is used to find an angle when you know the tangent of that angle. In mathematical notation, \text{Arctan} takes a number, say \( x \), and returns the angle \( \theta \) such that the tangent of \( \theta \) is equal to \( x \). This is written as \( \theta = \operatorname{Arctan}(x) \) where \( \tan(\theta) = x \).

• \( \operatorname{Arctan} \) is only defined when the tangent value is within the range all real numbers \((-\infty, \infty)\), but it returns angles within specific boundaries, usually from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) in radians.
• It helps in reversing the effect of the tangent function, allowing us to work backward from a tangent value to find the corresponding angle.

Understanding how \text{Arctan} works is essential when solving problems involving inverse trigonometric functions. It allows you to work with angles in their correct ranges, often converting results from degrees to radians, which are more commonly used in higher mathematics.

Tangent Function

The \text{tangent} function, often written as \( \tan \), is a fundamental trigonometric function. It relates the angle of a right triangle to the ratio of the opposite side over the adjacent side. Specifically, if \( \theta \) is an angle in a right triangle:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

This function is periodic, meaning it repeats its values in regular intervals, specifically every \( \pi \) radians.
Important characteristics include:

• It becomes undefined at certain points known as vertical asymptotes, which occur at \( \theta = \frac{(2n+1)\pi}{2} \) for all integers \( n \).
• It rapidly increases in value as the angle approaches its asymptotes.

The tangent function is widely used in solving practical problems and analyzing periodic phenomena in engineering and physics, where understanding angles and their size relationships is crucial.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which split a circle into 360 parts, radians relate directly to the circle's geometry:

• One full circle is \( 2\pi \) radians.
• A half-circle or straight angle is \( \pi \) radians.
• A quarter-circle, or right angle, measures \( \frac{\pi}{2} \) radians.

Radians provide a more natural way to express angles in mathematics, and they connect trigonometric functions seamlessly with calculus and complex numbers.
Converting between radians and degrees is simple:

• \( 1 \) radian = \( \frac{180}{\pi} \) degrees.
• \( 1 \) degree = \( \frac{\pi}{180} \) radians.

Using radians as a unit of measure is favored in scientific and engineering practices because it simplifies many formulas, particularly in calculus and physics, allowing for more precise calculations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables, usually within specific domain boundaries. These identities include relationships between functions like sine, cosine, and tangent.
Key identities include:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• \( 1 + \tan^2(\theta) = \sec^2(\theta) \)

These identities play a crucial role in simplifying trigonometric expressions and solving equations.

• They allow transformation and simplification of complex trigonometric problems.
• They form the foundation for calculus-based approaches to more complex problems, such as integration and differentiation involving trigonometric functions.

Understanding these identities allows one to navigate and solve trigonometric problems more effectively, providing the tools necessary to approach a wide range of mathematical applications.