Problem 45 - Precalculus Essentials

Question

In Exercises \(35-60\), find the reference angle for each angle. $$-150^{\circ}$$

Step-by-Step Solution

Answer

The reference angle for -150° is 30°.

Step 1: Determining the Quadrant

Identify the quadrant in which the angle is located. Since -150° is a clockwise rotation and greater than -180°, it lies in the second quadrant clockwise from the positive x-axis.

Step 2: Calculating the Reference Angle

The formula to find the reference angle in the second quadrant for a negative angle is (-180° - angle). Applying this formula, the reference angle becomes: (-180° - (-150°)) = 30°.

Key Concepts

Quadrants of a Circle

When dealing with angles, it's essential to understand the concept of quadrants. A full circle is divided into four quadrants, each representing a different range of angles. These quadrants help determine the position of an angle, which is particularly important when calculating the reference angle.

• The first quadrant encompasses angles from 0° to 90°, where both sine and cosine values are positive.
• The second quadrant covers angles from 90° to 180°, where sine values remain positive, while cosine values are negative.
• The third quadrant ranges from 180° to 270°, where both sine and cosine values are negative.
• The fourth quadrant spans from 270° to 360° (or 0°), where the sine values are negative and cosine values are positive.

Understanding the correct quadrant is crucial, specifically in our problem, where -150° falls into the second quadrant. This knowledge determines how we approach finding the reference angle.

Understanding Negative Angles

Negative angles can initially be perplexing, as they alter the angle's direction. Instead of measuring the angle counterclockwise from the positive x-axis, like positive angles, negative angles are gauged clockwise. This change in direction requires an understanding of how negative angles interact with the circle's quadrants.
Naturally, when an angle like -150° is introduced, it's crucial to first determine its position on the circle. By moving 150° clockwise, we can establish that it lies in the second quadrant's negative angle range. Realizing this helps avoid common errors when calculating the reference angle, ensuring the correct method and quadrant are applied.

Accurate Angle Measurement

Accurate measurement of angles involves more than just reading numbers; it demands an understanding of how angles are represented on the circle. An angle is the amount of rotation from the starting point, usually the positive x-axis, and it can be expressed in different forms like degrees or radians.
For our task, using degrees simplifies determining the reference angle. The reference angle is the acute angle that an angle makes with the x-axis, regardless of its position. It is always a positive angle, allowing us to find useful trigonometric functions' values.
In the case of -150°, once we know it lies in the second quadrant, we use the formula for the reference angle: \[ ext{Reference Angle} = 180° - | ext{Angle}| \]Thus, it simplifies to \[ 180° - 150° = 30° \].This value is crucial for calculations in trigonometry and ensures consistency, no matter whether dealing with positive or negative rotations.