Problem 50 - Precalculus

Question

For the following exercises, find the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal to the given angle. $$ -40^{\circ} $$

Step-by-Step Solution

Answer

The angle coterminal with \(-40^{\circ}\) is \(320^{\circ}\).

Step 1: Understanding Coterminal Angles

Two angles are coterminal if they share the same terminal side when in standard position. You can find a coterminal angle by adding or subtracting full rotations, which are multiples of \(360^{\circ}\).

Step 2: Find a Coterminal Angle

To find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(-40^{\circ}\), add \(360^{\circ}\) until the angle is positive. Thus, \(-40^{\circ} + 360^{\circ} = 320^{\circ}\).

Step 3: Verify the Result

Check that the angle \(320^{\circ}\) is between \(0^{\circ}\) and \(360^{\circ}\), and confirm it is coterminal with \(-40^{\circ}\). Since both criteria are met, \(320^{\circ}\) is the correct angle.

Key Concepts

Angle Measure

Angles are measured in degrees, and they represent the amount of rotation from one ray, known as the initial side, to another ray, called the terminal side. The entire circle is made up of 360 degrees. The angle measure helps us understand how much of a rotation is needed to reach from the starting position to the final position. Key things to remember about angle measures:

• A full rotation is 360 degrees.
• Half a circle is 180 degrees.
• A quarter turn is 90 degrees.

For example, if you rotate an object by 90 degrees, you will move it one-quarter way around a circle.

Standard Position

In mathematics, angles are often considered "in standard position". This means the angle is drawn on the coordinate plane. Its vertex is at the origin, and its initial side is along the positive x-axis. The terminal side is where it ends after rotation. Understanding the standard position is important because:

• It helps to consistently measure angles from the same starting point.
• Allows easier computation and comparison of angles.
• It aids in locating angles geometrically on the coordinate plane.

Thus, whenever you work with angles, positioning them in standard form makes mathematical operations more understandable and easier to follow.

Positive Angle

A positive angle in standard position is one that is generated by a counterclockwise rotation from the initial side to the terminal side. Most mathematical tools and geometry tasks consider positive angles by default. Remember about positive angles:

• They indicate counterclockwise (anti-clockwise) motion around a circle.
• They differ from negative angles, which rotate clockwise.
• A positive coterminal angle is often found by adding 360 degrees.

For instance, to find a positive angle coterminal with (-40^ {circ} ), we add (360^ {circ} ).

Angle Verification

Verifying angles is a crucial step when finding coterminal angles. It ensures that the resulting angle is both in the correct range and truly coterminal with the original angle. Here's how to verify an angle:

• Check that the angle lies within your given range, commonly between 0 and 360 degrees.
• Ensure the angle is coterminal, meaning it lands on the same terminal side as the original.
• In our task, verifying (320^ {circ} ) means checking it falls between 0 and 360 degrees and matching the terminal side of (-40^ {circ} ). Since all criteria match, the verification is successful.

Be thorough with this step to confirm your results are accurate and valid.