Problem 1 - Algebra 2 and Trigonometry

Question

Liam said that if \(0<\theta<90,\) then when the degree measure of a fourth- quadrant angle is \(-\theta,\) the degree measure of the reference angle is \(\theta .\) Do you agree with Liam? Explain why or why not.

Step-by-Step Solution

Answer

Yes, Liam is correct. The reference angle of \\( -\theta \\\) in the fourth quadrant is \\( \theta \\\).

Step 1: Understanding the Concept of Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle with the x-axis. For any angle in standard position, the reference angle is always between 0 and 90 degrees, regardless of the angle's location in a specific quadrant.

Step 2: Identifying the Fourth Quadrant

In the fourth quadrant, angles are measured clockwise from the positive x-axis, meaning they are negative when measured in standard position. An angle such as \( -\theta \) is located in the fourth quadrant when \( 0 < \theta < 90 \).

Step 3: Calculating the Reference Angle

For angles in the fourth quadrant, the reference angle \( \theta_{ref} \) is calculated as \( -(-\theta) = \theta \). This results because the terminal side of \( -\theta \) forms an \( \theta \) degree angle with the x-axis.

Step 4: Conclusion Based on Calculation

Since the calculation of the reference angle for \( -\theta \) results in \( \theta \), we can conclude that when the angle is in the fourth quadrant, the reference angle is indeed \( \theta \). Hence, Liam is correct.

Key Concepts

Fourth Quadrant

In the unit circle, the fourth quadrant is the region where the terminal side of an angle is located between \(270^\circ\) and \(360^\circ\).
This quadrant is found by moving clockwise from the positive x-axis. Angles in this region have negative degree measures when described in terms of standard position, as measured clockwise starting from \(0^\circ\).

A key characteristic of fourth quadrant angles is that their sine value is negative while their cosine value is positive.

• For example, an angle like \(-45^\circ\) or \(315^\circ\), which is equivalent to \(-45^\circ\) in standard position, lies in the fourth quadrant.
• This clockwise measurement results in angles having a negative sign.

The concept of negative and positive angles assists in understanding their position and behavior in coordinate systems.

Acute Angles

Acute angles are angles that measure less than \(90^\circ\). These angles are often used when determining a reference angle, especially because reference angles themselves must be acute.

Reference angles simplify complex angular measures and help in calculations involving trigonometric functions.

• For instance, if you have an angle \(130^\circ\), its reference angle would be \(50^\circ\) since you subtract its measure from \(180^\circ\) to find the acute angle it forms with the x-axis.
• The key takeaway is that no matter the quadrant where the original angle is situated, the reference angle remains an acute angle.

This consistent definition helps learners and users apply the concept across various problems and quadrants.

Terminal Side of an Angle

The terminal side of an angle is the ray or line that extends from the origin of an angle in standard position to define its direction or terminate at a coordinate. When drawing angles in standard position, the initial side lies on the positive x-axis.

The terminal side can tell you about the direction of rotation and the quadrant in which the angle lies.

• If an angle measures \(\theta > 0\), the terminal side rotates counterclockwise from the initial side.
• For a negative \(\theta\), the rotation is clockwise, which indicates angles like \(-\theta\) are often in quadrants like the fourth.

Understanding the terminal side is crucial, as it helps you identify the reference angle, which is the acute angle formed between the terminal side and the x-axis. This relation allows for easier interpretation of angle functions and improves comprehension of their geometric placements.