Problem 48 - 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. $$-159^{\circ}$$

Step-by-Step Solution

Answer

The angles are in the third quadrant; co-terminal angles are \(201^{\circ}\) and \(-519^{\circ}\).

Step 1: Understand Standard Position

An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis. We need to sketch the angle \(-159^{\circ}\). Since \(-159^{\circ}\) is negative, this means we will rotate clockwise from the positive x-axis.

Step 2: Sketch the Angle

Start at the positive x-axis and rotate clockwise through \(-159^{\circ}\). This rotation will land between the negative x-axis and the negative y-axis, specifically in the third quadrant.

Step 3: Find Positive Co-terminal Angle

An angle co-terminal with \(-159^{\circ}\) can be found by adding \(360^{\circ}\) to it. Thus, the positive co-terminal angle is:\[-159^{\circ} + 360^{\circ} = 201^{\circ}.\] The angle \(201^{\circ}\) is also in the third quadrant.

Step 4: Find Negative Co-terminal Angle

To find a negative co-terminal angle, you can subtract \(360^{\circ}\) from \(-159^{\circ}\). Thus, the negative co-terminal angle is:\[-159^{\circ} - 360^{\circ} = -519^{\circ}.\] The angle \(-519^{\circ}\) is also in the third quadrant.

Step 5: Identify Quadrant

The original angle \(-159^{\circ}\), along with its co-terminal angles \(201^{\circ}\) and \(-519^{\circ}\), all lie in the third quadrant.

Key Concepts

Co-terminal Angles

Co-terminal angles are angles that share the same terminal side when drawn in standard position. This means they look alike visually even if the amount of rotation differs. To find co-terminal angles, we commonly add or subtract full rotations (360 degrees) from the given angle.

Here's how it works:

• To find a positive co-terminal angle, add 360° to the given angle. For example, \(-159^{\circ} + 360^{\circ} = 201^{\circ}\), which is co-terminal with \(-159^{\circ}\).
• For a negative co-terminal angle, subtract 360°. Thus, \(-159^{\circ} - 360^{\circ} = -519^{\circ}\) is co-terminal.

This cycle can repeat infinitely, because every 360° rotation lands the terminal side back in the same position. These angles reside either in the same or in adjacent quadrants depending on the number of rotations involved.

Quadrants

The coordinate plane is divided into four quadrants. Each quadrant corresponds to a unique section. When dealing with angles, knowing which quadrant an angle falls into helps us understand its direction and orientation.

The quadrants are labeled counterclockwise starting from the positive x-axis, which are commonly referred to as:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative. This is where \(-159^{\circ}\) lands, as it rotates clockwise from the positive x-axis.
• Quadrant IV: x is positive, y is negative.

For the angle \(-159^{\circ}\), it lands in Quadrant III because it rotates clockwise through more than 90° from the positive x-axis.

Negative Angles

Negative angles rotate in the opposite direction of positive angles. In mathematics, the positive direction of an angle is counterclockwise, while negative angles move clockwise.

Consider the angle \(-159^{\circ}\). We start from the positive x-axis and move clockwise — hence, its location in Quadrant III. Using negative angles can help portray movements or rotations that move in the opposite direction of what we would usually expect from positive angles. Negative angles also allow us to find different perspectives of similar rotational positions.

Sketching Angles

Sketching angles helps visually understand their direction and position within the four quadrants.

To sketch \(-159^{\circ}\), follow these steps:

• Start at the positive x-axis, which is the usual place where angle measurement starts.
• Rotate clockwise by 159°. A full counterclockwise rotation is 360°, hence moving 159° clockwise effectively maps the direction.
• The terminal side will stop when the rotation completes, placing \(-159^{\circ}\) in the third quadrant, between the negative x-axis and negative y-axis.

Sketching can include marking the path the angle takes and drawing a curve to represent the angle's magnitude for clarity and understanding.