Problem 44 - 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. $$89^{\circ}$$

Step-by-Step Solution

Answer

89°, 449°, and -271° are all in Quadrant I.

Step 1: Sketching the Angle in Standard Position

To sketch the angle of \(89^{\circ}\) in standard position, start by drawing the positive x-axis as the initial side of the angle. Since \(89^{\circ}\) is slightly less than \(90^{\circ}\), draw the terminal side just a bit below the y-axis. This places the angle in Quadrant I.

Step 2: Drawing the Angle Notation

From the initial side on the positive x-axis, draw an arrow curving counterclockwise towards the terminal side that is positioned almost vertical, indicating a measure of \(89^{\circ}\). This visually represents the size of the angle in standard position.

Step 3: Finding a Positive Co-terminal Angle

To find a positive angle co-terminal with \(89^{\circ}\), add \(360^{\circ}\) to \(89^{\circ}\). The calculation gives \(89^{\circ} + 360^{\circ} = 449^{\circ}\). This positive angle also falls in the same quadrant as \(89^{\circ}\) when sketched.

Step 4: Finding a Negative Co-terminal Angle

To find a negative angle co-terminal with \(89^{\circ}\), subtract \(360^{\circ}\) from \(89^{\circ}\). The calculation gives \(89^{\circ} - 360^{\circ} = -271^{\circ}\). This negative angle also shares the same ending position and is in Quadrant I.

Step 5: Identifying Quadrants

The original angle \(89^{\circ}\) is in Quadrant I. The co-terminal angles \(449^{\circ}\) and \(-271^{\circ}\) are also in Quadrant I.

Key Concepts

Co-terminal Angles

Co-terminal angles are angles that share the same terminal side when drawn in standard position. They are essentially angles that "end" in the same spot. They differ from each other by whole multiples of a full circle, which is equivalent to 360 degrees.

• To find a co-terminal angle, you can add or subtract 360 degrees.
• For instance, with an angle of 89 degrees, adding 360 degrees results in a co-terminal angle of 449 degrees, while subtracting 360 degrees gives you -271 degrees.
• No matter how many times you add or subtract 360 degrees, the angle's terminal side will remain the same.

It's easy to find multiple co-terminal angles for a given angle, allowing for flexibility in calculations and understanding in different contexts. Co-terminal angles are useful for converting angles into measures that are easier to interpret or work with.

Quadrants of Angles

In the coordinate system, the circle is divided into four sections known as quadrants. These quadrants help in identifying the position of an angle’s terminal side. Each quadrant has distinct characteristics and determines the sign of the coordinate values (x and y).

• Quadrant I: Both x and y are positive. Angles here range from 0 to 90 degrees.
• Quadrant II: x is negative, y is positive. This quadrant covers angles from 90 to 180 degrees.
• Quadrant III: Both x and y are negative. It spans from 180 to 270 degrees.
• Quadrant IV: x is positive, y is negative, and it covers 270 to 360 degrees.

Knowing which quadrant an angle lies in helps in determining the angle’s properties, such as its direction and potential co-terminal angles. For example, an 89-degree angle, as depicted, lies in Quadrant I.

Positive and Negative Angles

Angles can be measured in both positive and negative directions. These directions signify the nature of the rotation from the initial side.

• Positive angles result from counterclockwise rotation from the initial side on the positive x-axis. An 89-degree angle is positive since its terminal side falls counterclockwise.
• Negative angles are indicated by a clockwise rotation, moving the terminal side in the opposite direction. The angle -271 degrees is a negative angle that results from this motion.

Understanding positive and negative angles not only clarifies the direction of rotation but also assists in solving and simplifying problems involving circular motion and periodic functions.