Problem 47 - Precalculus Essentials

Question

In Exercises \(35-60\), find the reference angle for each angle. $$-335^{\circ}$$

Step-by-Step Solution

Answer

The reference angle for the given angle \(-335^{\circ}\) is \(25^{\circ}\).

Step 1: Identify the Angle

The given angle is \(-335^{\circ}\). This is a negative angle, but the reference angle is always positive.

Step 2: Find the Equivalent Positive Angle

Since a full rotation (360 degrees) will bring us back to the same point, adding 360 degrees to any angle doesn't change its terminal side. Therefore, you can add 360 to the given angle and get an equivalent positive angle. So, \(-335^{\circ} + 360^{\circ} = 25^{\circ}\).

Step 3: Determine the Reference Angle

For angles that end in the first quadrant (like 25 degrees), the reference angle is just the angle itself. Therefore, the reference angle is \(25^{\circ}\).

Key Concepts

Negative Angles

Negative angles might seem confusing at first but are simple once understood. These angles are measured in the clockwise direction from the positive x-axis. Unlike positive angles that measure counterclockwise from the positive x-axis, negative angles move in the opposite direction. You can imagine a circle clock face here: instead of moving from 3 o'clock toward 12 o'clock (positive), you're moving toward 6 o'clock (negative).

• A negative angle helps in finding the direction differently, often used in navigation or physics.
• Negative angle measurements are handy for calculating equivalent positive angles by adding a full rotation of 360 degrees.

Considering the angle \(-335^{\circ}\), swing from 0 to -335 degrees on the circle, not the usual positive way. To simplify use, we convert it. Simply adding 360 degrees leads to the equivalent \(25^{\circ}\), making it easier to calculate reference angles.

Positive Angles

Positive angles are simple and intuitive. They rotate counterclockwise from the positive x-axis, covering ground we often naturally count. Think of moving from the 3 o'clock position along a clock face upwards towards 12 o'clock. This method is dominant in mathematical and geometric contexts.

• Positive angles are always greater than \(0\) but less than \(360^{\circ}\) for one complete rotation, considering practical uses.
• They simplify determining reference angles directly if they lie in the first quadrant.

Once derived from a negative angle by adding a full rotation (360 degrees), like from \(-335^{\circ}\) to \(25^{\circ}\), this resulting angle helps easier calculations and understanding within standard angle ranges.

Full Rotation

A full rotation in circular motion is defined as a complete circle, measured as \(360^{\circ}\). Whether dealing with positive or negative angles, this concept is crucial because it represents how angles repeat cyclically.

• Any angle plus or minus 360 degrees results in a similar position or direction, known as angular equivalency.
• Used especially in trigonometry when simplifying ambiguous angles into understandable or calculable results.

For instance, adding \(360^{\circ}\) to \(-335^{\circ}\) gives us \(25^{\circ}\), showing how one full rotation replicates original positions and greatly helps in determining reference angles or simplifying complex calculations.