Problem 10 - Basic Technical Mathematics with Calculus

Question

Determine one positive and one negative coterminal angle for each angle given. $$173^{\circ}$$

Step-by-Step Solution

Answer

Positive: \(533^{\circ}\), Negative: \(-187^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side. They can be found by adding or subtracting whole circles (360 degrees) to the given angle.

Step 2: Find a Positive Coterminal Angle

To find a positive coterminal angle, add 360 degrees to the given angle. For the given angle, add \(173^{\circ} + 360^{\circ}\).

Step 3: Calculation for Positive Angle

Perform the addition: \(173^{\circ} + 360^{\circ} = 533^{\circ}\). Thus, \(533^{\circ}\) is a positive coterminal angle.

Step 4: Find a Negative Coterminal Angle

To find a negative coterminal angle, subtract 360 degrees from the given angle. For the given angle, calculate \(173^{\circ} - 360^{\circ}\).

Step 5: Calculation for Negative Angle

Perform the subtraction: \(173^{\circ} - 360^{\circ} = -187^{\circ}\). Thus, \(-187^{\circ}\) is a negative coterminal angle.

Key Concepts

Positive Coterminal Angle

Understanding positive coterminal angles is quite straightforward. An angle can have multiple angles that end at the same spot on a circle, known as coterminal angles. To find a positive coterminal angle, we add 360 degrees, or one full circle rotation, to the original angle.

For example, let's consider an original angle of \(173^{\circ}\). By adding \(360^{\circ}\), the equation becomes \(173^{\circ} + 360^{\circ} = 533^{\circ}\). Thus, \(533^{\circ}\) is a positive coterminal angle of \(173^{\circ}\).

This method works because adding full circle rotations always brings you back to the same terminal position, thereby creating a coterminal angle.

• Positive angles are greater than the original angle.
• The process can be repeated to find more positive coterminal angles.

Negative Coterminal Angle

To find a negative coterminal angle, we need to understand a slightly different approach. This involves subtracting \(360^{\circ}\) from the original angle to result in an angle with the same terminal location but going in the reverse direction on the circle.

For the angle \(173^{\circ}\), subtract \(360^{\circ}\). The calculation is \(173^{\circ} - 360^{\circ} = -187^{\circ}\). Thus, \(-187^{\circ}\) is a negative coterminal angle for \(173^{\circ}\).

This process shows how something as simple as subtraction can lead us to discover different angles that visually end up in the same spot.

• Negative angles indicate a clockwise direction on the circle.
• Subtracting \(360^{\circ}\) can be repeated for additional negative angles.

Angle Addition and Subtraction

The principles of angle addition and subtraction are foundational to understanding coterminal angles. These operations help us navigate the circular nature of angles effortlessly.

Adding \(360^{\circ}\) to an angle allows finding further positive angles, taking you through several full rotations. Conversely, subtracting \(360^{\circ}\) will shift the angle into the negative direction, reflecting a clockwise movement.

The essence here is simple: every full rotation (\(360^{\circ}\)) does not alter the position of the terminal side, only the numeric value of the angle changes. This is why both methods generate coterminal angles.

• Ensure you perform arithmetic operations carefully.
• Remember, direction doesn't change the terminal position.

Understanding these concepts allows you to explore a variety of situations using angle calculations confidently.