Problem 321 - Precalculus

Question

For the following exercises, draw the angle provided in standard position on the Cartesian plane. $$ -210^{\circ} $$

Step-by-Step Solution

Answer

Draw the angle in the second quadrant, as it's coterminal with \(150^{\circ}\).

Step 1: Understanding the Angle

The angle given is \(-210^{\circ}\). Since it is negative, it represents a rotation in the clockwise direction.

Step 2: Finding Coterminal Angle

To make drawing easier, convert \(-210^{\circ}\) to a positive coterminal angle by adding \(360^{\circ}\), the full revolution. \(-210^{\circ} + 360^{\circ} = 150^{\circ}\). This means \(-210^{\circ}\) is coterminal with \(150^{\circ}\).

Step 3: Drawing the Positive Coterminal Angle

Start from the positive x-axis and rotate counterclockwise to draw \(150^{\circ}\). Since \(150^{\circ}\) is less than \(180^{\circ}\), it lands in the second quadrant.

Step 4: Finalizing the Angle in Standard Position

Using \(-210^{\circ}\) and knowing it's equivalent to \(150^{\circ}\), draw a clockwise rotation starting from the positive x-axis reaching the second quadrant.

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position on the Cartesian plane. In simpler terms, they "end" at the same place but may have had different journeys getting there. To determine coterminal angles, you can add or subtract multiples of 360° to your given angle. This is because 360° represents one complete revolution around a circle.

For example, in the exercise,

• The original angle is \(-210^{\circ}\).
• Add 360° to find a coterminal angle in a positive direction: \(-210^{\circ} + 360^{\circ} = 150^{\circ}\).

Knowing how to find coterminal angles is useful as it allows us to deal with positive angles even if the original angle given is negative.

When sketching or understanding angles, coterminal angles help by simplifying complex angle measures and allowing easier visualization.

Negative Angles

Negative angles are angles that represent a clockwise direction of rotation rather than the more common counterclockwise direction used for positive angles. In the Cartesian plane, this doesn't mean the angle is "less" or "bad," it just starts in a different rotational direction.

In the exercise:

• The specified angle is \(-210^{\circ}\).
• This indicates a rotation of 210 degrees in a clockwise direction starting from the positive x-axis.

Understanding the concept of negative angles allows for better interpretation and drawing, especially when it comes to distinguishing between the direction and position of angles.

By converting negative angles to their positive coterminal counterparts, as shown in the solution, you can make visualization easier and recognize equivalent positions on a standard grid.

Cartesian Plane

The Cartesian plane is a two-dimensional graph with a horizontal axis, known as the x-axis, and a vertical axis, known as the y-axis. It provides a grid for plotting points, lines, and shapes, and is instrumental in graphing concepts like angles in standard position.

In the context of angles:

• Angles originate from the intersection of the x and y axes, known as the origin.
• An angle in standard position starts from the positive x-axis.

In the exercise, converting \(-210^{\circ}\) to a positive angle like \(150^{\circ}\) allows for easier plotting on the Cartesian plane. This visualization ability is crucial for accurately drawing angles and understanding their spatial relationships within the plane.

The Cartesian plane, with its grid system, enables clarity in demonstrating where an angle lies and how rotations affect their positioning.

Quadrants

Quadrants divide the Cartesian plane into four distinct areas, each designated by a Roman numeral. Understanding which quadrant an angle falls into helps in graphically representing the angle and predicting the behavior of equations or functions related to it.

Here's how the quadrants are arranged:

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

The exercise involves a coterminal angle of \(150^{\circ}\), which lies in the second quadrant. This quadrant indicates that angles have started a counterclockwise rotation but have not completed one full half-circle, thus ending around 150° or an equivalent negative angle.

Knowing which quadrant an angle settles into helps identify the signs of coordinates and the potential impact on related functions or graphing, offering a systematic approach to tackle similar angle-related problems.