Problem 2 - Algebra 2

Question

Solve each equation by finding the value of \(x\) to the nearest degree. \(\operatorname{Arctan} 0=x\)

Step-by-Step Solution

Answer

The value of \(x\) is \(0^\circ\).

Step 1: Understand the Arctan Function

The arctan function, also known as the inverse tangent function, is used to find an angle whose tangent is a given number. In this problem, we are given \( \tan(x) = 0 \). We are asked to find the angle \( x \) that satisfies this relation.

Step 2: Apply the Tangent Function Property

Recall that the tangent of an angle is 0 when the angle itself is 0 degrees (or 0 radians), because \( \tan(0) = 0 \). Therefore, the arctan of 0 is 0, since that is the angle that yields a tangent of 0.

Step 3: Conclusion and Nearest Degree

The arctan of a number can yield multiple angles, particularly when considering additional rotations. However, when considering the principal value, which is between \(-90^\circ\) and \(90^\circ\), the angle that satisfies \( \operatorname{Arctan} 0 = x \) is \(x = 0^\circ\). As we are asked for the nearest degree, the answer remains \(0^\circ\).

Key Concepts

Angle Measurement

When dealing with trigonometric functions, angle measurement is pivotal. Angles can be measured in degrees or radians, depending on the context. For instance, in many educational settings, degrees are more commonly used. These measurements help us to determine positions and orientations in a plane.

• Degrees: The most familiar measurement unit for angles. There are 360 degrees in a complete circle. Each degree can be subdivided further into minutes and seconds.


• Radians: Another unit of measurement for angles. Here, the angle is determined based on the radius of a circle. In one full circle, there are approximately 6.28 radians (or exactly 2π). Radians are often used in higher-level mathematics due to their natural relation with the circle's properties.

Understanding angle measurement is crucial as it dictates how inverse trigonometric functions, like arctan, interpret and provide results. Whether dealing with degrees or radians, being precise with those measurements ensures clarity and accuracy in results.

Tangent Function

The tangent function is a key concept in trigonometry. It relates the angles of a right triangle to the ratios of its sides. Specifically, for an angle θ:
\[ \tan(θ) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\]

• Definition: The tangent function is often represented as \( \tan(θ) \). It gives the ratio of the side opposite the angle to the adjacent side in a right triangle.


• Properties: The function has a periodic nature with a period of 180 degrees (or π radians). This means \( \tan(θ) = \tan(θ + 180°) \).
  At 0 degrees, \( \tan(0) = 0 \), meaning that the ratio of the opposite side to the adjacent side is zero.


• Graph: The graph of the tangent function shows that it rises and falls repeatedly across the x-axis. At certain points, called asymptotes (e.g., 90°, 270°, etc.), the function is undefined.

The tangent function's behavior and properties make it invaluable, especially when paired with its inverse in solving problems such as finding angles when the tangent value is known.

Principal Value

The concept of the principal value is essential when dealing with inverse trigonometric functions. For arctan (or inverse tangent), principal values provide a specific range for angle outputs, ensuring consistency and clarity.

• Principal Value Range: For the arctan function, principal values are restricted between \(-90°\) and \(90°\). This ensures that for any real number, the arctan function yields only one value.


• Why Use Principal Values? Due to the periodic nature of trigonometric functions, many angles can yield the same tangent. Principal values help select one specific angle, usually the smallest one in magnitude.

In the example where \( \operatorname{Arctan} 0 = x \), the principal value provides clarity in identifying that \(x = 0°\) is the correct angle. It prevents confusion from multiple solutions and focuses on the primary solution commonly used in calculations.