Problem 47 - 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. $$-61^{\circ}$$

Step-by-Step Solution

Answer

-61°, 299°, and -421° are all in the fourth quadrant.

Step 1: Sketch the Angle in Standard Position

A standard position angle has its initial side on the positive x-axis. Starting from here, rotate clockwise by 61 degrees to represent the angle \(-61^{\circ}\). Draw an arrow indicating the rotation from the positive x-axis in a clockwise direction. The terminal side will be in the fourth quadrant.

Step 2: Calculate Positive Co-Terminal Angle

To find a positive co-terminal angle with \(-61^{\circ}\), add \(360^{\circ}\) to \(-61^{\circ}\): \[ -61^{\circ} + 360^{\circ} = 299^{\circ} \] So, \(299^{\circ}\) is a positive angle co-terminal with \(-61^{\circ}\) and is also in the fourth quadrant.

Step 3: Calculate Negative Co-Terminal Angle

To find a negative co-terminal angle with \(-61^{\circ}\), subtract \(360^{\circ}\) from \(-61^{\circ}\): \[ -61^{\circ} - 360^{\circ} = -421^{\circ} \]Thus, \(-421^{\circ}\) is a negative angle co-terminal with \(-61^{\circ}\) and is also in the fourth quadrant.

Step 4: Identify Quadrant for Each Angle

The angles \(-61^{\circ}\), \(299^{\circ}\), and \(-421^{\circ}\) all have their terminal sides in the fourth quadrant. Any angle between \(270^{\circ}\) and \(360^{\circ}\) or \(-90^{\circ}\) and \(0^{\circ}\) ideally ends in the fourth quadrant.

Key Concepts

Angles

Angles are fundamental concepts in trigonometry and geometry. They are formed by two rays with a common endpoint known as the vertex. The amount of rotation between the initial position (usually on the positive x-axis) and the final position of a ray determines the measure of an angle.
Angles can be measured in degrees or radians, and they can be positive or negative based on their direction of rotation. Positive angles are measured counterclockwise from the initial side, while negative angles are measured clockwise.
Understanding angles is crucial for grasping more complex trigonometric concepts.

Standard Position

When discussing angles in trigonometry, it's important to know about the standard position. An angle is said to be in standard position when its vertex is at the origin of a coordinate system, and its initial side lies along the positive x-axis.
This serves as a starting point for measuring angles, making it easier to identify their rotation direction and terminal side location.
By establishing a common frame of reference, standard position helps in analyzing and solving problems involving angles and their relationships in a consistent manner.

Co-Terminal Angles

Co-terminal angles are angles that share the same terminal side when drawn in standard position. In other words, despite having different measures, their terminal sides overlap.
To find co-terminal angles, you can keep adding or subtracting full rotations, which is usually 360 degrees or \(2\pi\) radians in terms of degrees or radians, respectively.
For instance, to find a positive co-terminal angle for \(-61^{\circ}\), you add 360 degrees to get \(299^{\circ}\). Similarly, subtracting 360 degrees gives a new co-terminal angle, \(-421^{\circ}\).
Identifying co-terminal angles aids in exploring different angle scenarios while maintaining the same geometrical orientation.

Quadrants

In the coordinate plane, quadrants are divided into four regions, each labeled from I to IV, moving counterclockwise from the positive x-axis:

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

Knowing which quadrant an angle's terminal side lies in is critical for understanding its sign and properties.
For example, angles like \(-61^{\circ}\), \(299^{\circ}\), and \(-421^{\circ}\) all end in Quadrant IV, which indicates that they have a positive x and a negative y-coordinate.
Using quadrant information helps in determining trigonometric values and understanding angles' behaviors in different scenarios.