Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Sketch each angle in standard position. Draw an arrow representing the correct amount of notation. Find the measure of two other angles, one positive and one negative, that are co terminal with the given angle. Give the quadrant of each angle. $$234^{\circ}$$

Step-by-Step Solution

Answer

$234^\circ$ is in the third quadrant with co-terminal angles $594^\circ$ (positive) and $-126^\circ$ (negative).

Step 1: Understand Angle in Standard Position

An angle is in standard position if its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis. We need to sketch the angle \(234^\circ\) by rotating the terminal side counterclockwise from the initial side.

Step 2: Sketch the Angle $234^\circ$

Starting from the positive x-axis, move counterclockwise. Since \(234^\circ\) is more than \(180^\circ\) but less than \(270^\circ\), the terminal side of the angle is located in the third quadrant. Draw an arrow showing the rotation to indicate the degree measure.

Step 3: Find a Positive Co-terminal Angle

To find a positive co-terminal angle, add \(360^\circ\) to \(234^\circ\). This gives: \(234^\circ + 360^\circ = 594^\circ\). Thus, \(594^\circ\) is a positive angle co-terminal with \(234^\circ\).

Step 4: Find a Negative Co-terminal Angle

To find a negative co-terminal angle, subtract \(360^\circ\) from \(234^\circ\). This gives: \(234^\circ - 360^\circ = -126^\circ\). Thus, \(-126^\circ\) is a negative angle co-terminal with \(234^\circ\).

Step 5: Identify the Quadrants

The angle \(234^\circ\) is in the third quadrant. The co-terminal angles \(594^\circ\) and \(-126^\circ\) are also in the third quadrant, as co-terminal angles share the same terminal side.

Key Concepts

Standard Position

Understanding how an angle is positioned in a coordinate plane is crucial. In standard position, the angle's vertex is at the origin of the coordinate plane, and its initial side is on the positive x-axis. From this starting point, the terminal side tells us where the angle finishes after a certain amount of rotation. For example, when sketching the angle \(234^\circ\), imagine beginning on the positive x-axis and rotating the terminal side counterclockwise until you reach the desired measurement. This concept helps in visualizing angles and their correct positions, which is essential for understanding co-terminal angles.

Quadrants

The coordinate plane is divided into four quadrants, which helps us determine an angle's final position after rotation. The quadrants are numbered counterclockwise starting from the positive x-axis. Here's a simple view:

• Quadrant I: Positive x and y coordinates.
• Quadrant II: Negative x and positive y coordinates.
• Quadrant III: Negative x and y coordinates.
• Quadrant IV: Positive x and negative y coordinates.

An angle like \(234^\circ\) falls in Quadrant III since it is more than \(180^\circ\) but less than \(270^\circ\). Identifying the quadrant is important for providing context about the angle.

Angle Sketching

Sketching an angle involves visualizing its position on the coordinate plane. To sketch \(234^\circ\), start from the positive x-axis—this is the initial side. Then move in a counterclockwise direction until the angle term completes at \(234^\circ\). You'll notice the angle will end in the third quadrant of the coordinate plane. Drawing an arrow along the path of rotation helps in displaying the magnitude and direction of the angle. This exercise aids in both understanding the angle's magnitude and its position, which is foundational for grasping more complex trigonometric concepts.

Positive and Negative Angles

Angles can be expressed as both positive and negative to describe their rotations. A positive angle is measured counterclockwise from the initial side, while a negative angle is measured clockwise. For instance, \(234^\circ\) can be co-terminal with a positive angle like \(594^\circ\) by adding \(360^\circ\). Similarly, by subtracting \(360^\circ\), you get a negative co-terminal angle, \(-126^\circ\). Co-terminal angles share the same terminal side on the coordinate plane, reinforcing that the angle's position depends on its terminal side, not the path taken.