Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Give the reference angle for each angle measure. $$-135^{\circ}$$

Step-by-Step Solution

Answer

The reference angle for \(-135^{\circ}\) is \(45^{\circ}\).

Step 1: Understanding Reference Angles

The reference angle is the smallest angle a given angle can make with the x-axis. It is always positive and is between 0° and 90°.

Step 2: Convert Negative Angle to Positive

Since we are given a negative angle of \(-135^{\circ}\), we need to determine its equivalent positive angle by adding \(360^{\circ}\). Thus, \(-135^{\circ} + 360^{\circ} = 225^{\circ}\).

Step 3: Find Quadrant and Reference Angle Formula

The angle \(225^{\circ}\) is in the third quadrant. In this quadrant, the reference angle is calculated as \( heta - 180^{\circ}\).

Step 4: Calculate the Reference Angle

For \(225^{\circ}\), calculate the reference angle: \(225^{\circ} - 180^{\circ} = 45^{\circ}\).

Key Concepts

Negative Angles

Angles are typically measured from the positive x-axis in a counterclockwise direction. When an angle is expressed as a negative value, it indicates a clockwise rotation. Negative angles might seem confusing initially, but they are just as valid and can be converted for easier understanding.
To convert a negative angle into a positive one that is equivalent, you simply add 360°, as a full circle is 360 degrees. For example, converting \(-135^{\circ}\):

• Add 360° to \(-135^{\circ}\), resulting in \(225^{\circ}\).

This system of conversion allows us to express angles in a more familiar form, making it easier to determine their properties, such as reference angle and quadrant.

Quadrants

Understanding which quadrant an angle falls into is crucial for calculating its reference angle. The coordinate plane is divided into four quadrants which are numbered counterclockwise starting from the positive x-axis.
Each quadrant has different rules for calculating the reference angle:

• First Quadrant: Angle between 0° to 90°. The reference angle is the angle itself.
• Second Quadrant: Angle between 90° to 180°. Subtract the angle from 180° to find the reference angle.
• Third Quadrant: Angle between 180° to 270°. Subtract 180° from the angle to find the reference angle.
• Fourth Quadrant: Angle between 270° to 360°. Subtract the angle from 360° to find the reference angle.

For an angle like \(225^{\circ}\), it falls in the third quadrant, where the reference angle is determined by subtracting 180° from the angle, yielding a reference angle of \(45^{\circ}\). This method makes it simple to work with both positive and negative angles by placing them within the standard system of quadrants.

Angle Conversion

Converting an angle into a reference angle helps simplify calculations in trigonometry and geometry. This conversion process involves different steps but ensures you always end up with an angle between 0° and 90°. To find a reference angle:

• First, ensure the angle is expressed in positive degrees by adding or subtracting 360° as needed, so it's within a 0° to 360° range.
• Next, determine the quadrant in which the angle lies to apply the correct formula for the reference angle.
• Apply the formula for that quadrant to find the reference angle, simplifying calculations and understanding.

This conversion makes it easier to analyze the behavior of angles in mathematical contexts. For example, trigonometric functions often rely on reference angles since it helps ascertain their values easily by referencing them to the primary axis.