Problem 38 - Algebra 2

Question

Sketch each angle. Then find its reference angle. \(-210^{\circ}\)

Step-by-Step Solution

Answer

The reference angle for -210° is 30°.

Step 1: Understand the Concept of Reference Angle

The reference angle for any angle is the smallest positive angle between the terminal side of the given angle and the x-axis. It is always measured in positive degrees.

Step 2: Sketch the Angle -210°

Since -210° is negative, we will move clockwise from the positive x-axis. A full clockwise rotation is 360°, so after moving -210°, we will land 210° on the clockwise direction or equivalently 150° on the counterclockwise direction from the positive x-axis.

Step 3: Determine the Quadrant

The angle -210° places the terminal side in the second quadrant because it is equivalent to moving 150° counterclockwise from the x-axis.

Step 4: Calculate the Reference Angle

In the second quadrant, the reference angle is calculated as 180° minus the angle's positive version in degrees from the x-axis. Thus, the reference angle is 180° - 150° = 30°.

Key Concepts

Angle Sketching

Sketching angles is an essential skill in trigonometry. It allows you to visualize where an angle lands on the coordinate plane. This process is straightforward when you follow a few simple steps.

• Start from the positive x-axis. This is your starting point, or your 'zero' degrees.
• If your angle is positive, rotate counterclockwise. If it is negative, rotate clockwise.
• Proceed by counting the degrees according to your angle's measure.
• Continue until you reach the desired angle. Make sure to note the direction you are moving in.

In our exercise, the angle is \(-210^{\circ}\), which means we rotate clockwise starting from the positive x-axis. Visualizing this helps us see where the terminal side of the angle lies, making it easier to find the reference angle later on. Remember that sketching is like a map; it guides your understanding and shows precisely where angles position themselves within the coordinate plane.

Negative Angles

Angles can be positive or negative, a concept that might be tricky at first. Positive angles increase from the positive x-axis in a counterclockwise direction, while negative angles decrease in a clockwise direction. Understanding this concept is crucial for sketching and analyzing angles.

For example, when we have \(-210^{\circ}\), we rotate from the positive x-axis moving clockwise. A complete circle is \(360^{\circ}\); this means that \(-210^{\circ}\) takes us beyond halfway around the circle. This notion sometimes needs practice, but once grasped, makes working with angles in trigonometry much easier. Negative angles are simply another way of describing direction and magnitude, and recognizing them aids in finding the angle's position accurately.

Quadrants in Trigonometry

The plane is divided into four quadrants, each a section formed by the positive and negative x and y axes. Each quadrant represents a different combination of positive and negative x and y coordinates. Understanding which quadrant an angle lies in helps determine its trigonometric values and reference angle.

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

For \(-210^{\circ}\), when we rotate 210 degrees clockwise from the positive x-axis, we land in the second quadrant. Since the terminal side has traveled from a negative x coordinate to a positive y coordinate direction, it is important to know this as it affects how we calculate the reference angle. In this case, given our angle lands in Quadrant II, we compute the reference angle by subtracting from \(180^{\circ}\), highlighting how quadrant determination simplifies trigonometric analysis.