Problem 3 - Precalculus Mathematics for Calculus

Question

Find the reference angle for the given angle. (a) \(150^{\circ}\) (b) \(330^{\circ}\) (c) \(-30^{\circ}\)

Step-by-Step Solution

Answer

(a) 30° (b) 30° (c) 30°

Step 1: Understanding Reference Angles

A reference angle is the positive acute angle that the terminal side of a given angle makes with the x-axis. The reference angle is always between 0 and 90 degrees.

Step 2: Reference Angle for Positive Angles in Quadrant II

For angles in the second quadrant (90° to 180°), the reference angle can be found by subtracting the angle from 180°. Calculate the reference angle for 150°:\[180^{\circ} - 150^{\circ} = 30^{\circ}\]Thus, the reference angle for 150° is 30°.

Step 3: Reference Angle for Positive Angles in Quadrant IV

For angles in the fourth quadrant (270° to 360°), the reference angle is found by subtracting 360° from the angle.Calculate the reference angle for 330°:\[360^{\circ} - 330^{\circ} = 30^{\circ}\]Therefore, the reference angle for 330° is 30°.

Step 4: Reference Angle for Negative Angles

For negative angles, first convert the angle to a positive angle by adding 360° (if necessary), then determine the reference angle.Calculate the equivalent positive angle for -30°:\[360^{\circ} + (-30^{\circ}) = 330^{\circ}\]Since 330° is in the fourth quadrant, use the method described in Step 3 to find its reference angle:\[360^{\circ} - 330^{\circ} = 30^{\circ}\]Thus, the reference angle for -30° is 30°.

Key Concepts

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants, which help in determining the sign and reference angle of angles measured in degrees.
Each quadrant represents a different range of angles:

• First Quadrant: 0° to 90°.
• Second Quadrant: 90° to 180°.
• Third Quadrant: 180° to 270°.
• Fourth Quadrant: 270° to 360°.

The quadrants help us in finding the reference angle because each angle from various quadrants relates to an acute angle. By knowing which quadrant an angle lies in, we can apply different calculations to find its corresponding reference angle.
For instance:

• In the Second Quadrant, the reference angle is found by subtracting the angle from 180°.
• In the Fourth Quadrant, subtract the angle from 360° to find the reference angle.

Recognizing the quadrant allows students to use the appropriate formula to quickly find the reference angle.

Angles and Degrees

Angles can be understood easily when expressed in degrees. It’s essential to comprehend how different angles influence calculations involving trigonometry.
Degrees are a measure of rotation, with a complete circle corresponding to 360°. Each angle can be positive or negative:

• Positive Angles: Angles measured counterclockwise from the positive x-axis.
• Negative Angles: Angles measured clockwise from the positive x-axis.

Understanding angles in degrees helps when determining the reference angle, no matter the quadrant. Positive angles like 150° and 330°, or negative angles like -30°, may appear complex at first.
Transforming negative angles into positive equivalents by adding 360° simplifies the process.
Knowing this makes it easier to find reference angles and consequently solve trigonometric problems.

Acute Angles

An acute angle is one of the fundamental types in geometry, and trigonometry also makes use of this concept.
Acute angles measure less than 90°, making them ideal for reference angles since they represent the smallest angle between the terminal side of an angle and the x-axis. This makes them valuable for simplifying trigonometric calculations.
Every angle, whether it lies in the first, second, third, or fourth quadrant, can ultimately be converted to an acute angle.
Examples include:

• An angle like 150°, from the second quadrant, sees its reference expressed as 30°, which is acute.
• -30° is converted to 330° (in the fourth quadrant), also leading to a reference angle of 30°.

By ensuring we convert all angles into acute angles (between 0° and 90°), we make calculations simpler and more straightforward, aiding in learning and understanding trigonometry.