Problem 13 - Algebra 2 and Trigonometry

Question

In \(8-17,\) for each angle with the given degree measure, find the measure of the reference angle. \(310^{\circ}\)

Step-by-Step Solution

Answer

The reference angle is \(50^\circ\).

Step 1: Understand the concept of a reference angle

A reference angle is the smallest angle between the terminal side of the given angle and the x-axis. Reference angles are always between \(0^\circ\) and \(90^\circ\).

Step 2: Determine the quadrant of the given angle

Since the angle is \(310^\circ\), it falls in the fourth quadrant, because angles between \(270^\circ\) and \(360^\circ\) are located in the fourth quadrant.

Step 3: Calculate the reference angle for a fourth quadrant angle

In the fourth quadrant, the reference angle is given by \(360^\circ - \theta\), where \(\theta\) is the given angle. In this case, calculate \(360^\circ - 310^\circ\).

Step 4: Perform the subtraction

Subtract \(310^\circ\) from \(360^\circ\) to find the reference angle: \[360^\circ - 310^\circ = 50^\circ\].Thus, the reference angle is \(50^\circ\).

Key Concepts

Fourth Quadrant

The concept of quadrants is essential to understanding angles and their properties. Imagine a circle divided into four parts, each known as a quadrant, using two perpendicular lines crossing at the origin. The fourth quadrant is the part where angles range from beyond \(270^\circ\) up to \(360^\circ\). In this section, these angles are often negative in Tangent and sine values.

When you have an angle like \(310^{\circ}\), it fits in the fourth quadrant. This quadrant helps determine how we calculate the reference angle, which is crucial for trigonometric functions and other mathematical calculations. Just remember: each quadrant helps us figure out how to process angles and their properties effectively.

Angle Measurement

Understanding angle measurement is crucial for figuring out angles and their positions on a coordinate plane. Angles are measured in degrees, a way of dividing a circle into 360 equal parts. This measure helps in expressing angles as numerical values.

• An angle of \(0^{\circ}\) begins on the positive x-axis.
• A full circle measures \(360^{\circ}\).
• To measure past \(360^{\circ}\), just keep adding to make larger angles.

Angles can help us understand position and rotation. With an angle of \(310^{\circ}\), it lets us know it's beyond the third quadrant and into the fourth. This measurement not only tells us about an angle's size but also its location relative to the axes.

Subtraction in Degrees

To find reference angles in the fourth quadrant, subtraction in degrees becomes very important. The numbers tell us how far the angle is from completing a full circle of \(360^{\circ}\). To figure out the reference angle when the given angle is in the fourth quadrant, you need to subtract the angle from \(360^{\circ}\).

In this specific situation with \(310^{\circ}\), subtracting gives us:

• \(360^{\circ} - 310^{\circ} = 50^{\circ}\).

This result of \(50^{\circ}\) is the reference angle, lying between \(0^{\circ}\) and \(90^{\circ}\). Remember, the reference angle is always a positive acute angle, helping in simplifying calculations for trigonometric functions. By understanding this subtraction method, you can confidently find reference angles for any angle in the fourth quadrant.