Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Give the reference angle for each angle measure. $$-60^{\circ}$$

Step-by-Step Solution

Answer

The reference angle for \(-60^{\circ}\) is \(60^{\circ}\).

Step 1: Understanding Reference Angles

A reference angle is the acute angle a given angle makes with the x-axis on the coordinate plane. It is always between \(0^{\circ}\) and \(90^{\circ}\), irrespective of the original angle's sign or quadrant.

Step 2: Normalize the Angle into Standard Position

Normalize the given angle \(-60^{\circ}\) by adding \(360^{\circ}\) to find its equivalent in the standard position. Calculate: \(-60^{\circ} + 360^{\circ} = 300^{\circ}\).

Step 3: Determine the Quadrant of the Standard Position Angle

The angle \(300^{\circ}\) lies in the fourth quadrant, where angles range between \(270^{\circ}\) and \(360^{\circ}\).

Step 4: Calculate the Reference Angle for a Fourth Quadrant Angle

For angles in the fourth quadrant, the reference angle is found by subtracting the angle from \(360^{\circ}\). So, calculate: \(360^{\circ} - 300^{\circ} = 60^{\circ}\).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is uniquely specified by a pair of numerical coordinates. This system is composed of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). When it comes to angles, this plane becomes an invaluable tool for visualizing and describing their positions and properties.
The intersection of the x and y axes creates four sections, known as quadrants, where angles can reside. Each quadrant has distinct characteristics:

• The first quadrant includes angles from 0° to 90°.
• The second quadrant spans from 90° to 180°.
• The third quadrant ranges from 180° to 270°.
• The fourth quadrant covers angles from 270° to 360°.

Reference angles are related to these quadrants, as they represent the "smaller" angle an initial angle makes with the x-axis. On the coordinate plane, understanding where an angle lies helps in identifying this reference angle and aids in simplifying trigonometric calculations.

Standard Position

An angle is said to be in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis. This standardization makes it easier to locate angles and compute related measures, such as reference angles.
For angles not initially in standard position, such as time and negative angles, normalization is key. This involves adjusting the angle by adding or subtracting full circles (360°) to bring it between 0° and 360°. For instance, \(-60^{\circ} + 360^{\circ} = 300^{\circ}\), converts a negative angle into a positive one, placing it in a measurable and accessible position.
Knowing standard position helps you determine which quadrant the angle resides in, which in turn, assists in calculations like finding reference angles. These calculations simplify and standardize operations with angles.

Fourth Quadrant

The fourth quadrant of the coordinate plane is where angles ranging from 270° to 360° are located. This quadrant is characterized by having angles whose terminal side lies in the lower-right part of the plane.
In the fourth quadrant,

• The x-values of points are positive.
• The y-values of points are negative.

When dealing with an angle in the fourth quadrant, calculating the reference angle involves subtracting the given angle from 360°. This gives the acute angle it forms with the x-axis. For example, an angle of \(300^{\circ}\)in the fourth quadrant results in a reference angle of \(360^{\circ} - 300^{\circ} = 60^{\circ}\).
Understanding the properties of the fourth quadrant helps in determining angle qualities quickly, especially when combined with trigonometric functions that depend on quadrant location.