Problem 43 - Precalculus Mathematics for Calculus

Question

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$-800^{\circ}$$

Step-by-Step Solution

Answer

The angle \(280^{\circ}\) is coterminal with \(-800^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find a coterminal angle within a range, such as between \(0^{\circ}\) and \(360^{\circ}\), we need to add or subtract full rotations (multiples of \(360^{\circ}\)) from the given angle.

Step 2: Calculate the Coterminal Angle

Start with the given angle \(-800^{\circ}\). To transform this angle to one between \(0^{\circ}\) and \(360^{\circ}\), continue adding \(360^{\circ}\) until the result is within the desired range.

Step-by-Step Calculation

First, add \(360^{\circ}\) to \(-800^{\circ}\): \[-800^{\circ} + 360^{\circ} = -440^{\circ}\]Since \(-440^{\circ}\) is still not within the range, add another \(360^{\circ}\): \[-440^{\circ} + 360^{\circ} = -80^{\circ}\]Again, add \(360^{\circ}\): \[-80^{\circ} + 360^{\circ} = 280^{\circ}\]Now, \(280^{\circ}\) is between \(0^{\circ}\) and \(360^{\circ}\).

Step 4: Verification

Verify the calculation by checking if \(280^{\circ}\) is indeed coterminal with \(-800^{\circ}\):Subtract \(800^{\circ}\) from \(280^{\circ}\): \[280^{\circ} - 800^{\circ} = -520^{\circ}\]\(-520^{\circ}\) is divisible by \(360^{\circ}\), confirming \(280^{\circ}\) as a coterminal angle.

Key Concepts

Angle Conversion

When we talk about angle conversion, we're referring to the process of transforming an angle measured in degrees into an equivalent angle that fits within a specific range, often between 0° and 360°. This is particularly useful when you're dealing with angles that fall outside this range, whether they're negative or greater than 360°.
Understanding how to convert angles is crucial in problems involving coterminal angles. Coterminal angles are found by adding or subtracting multiples of a full circle (360°) to the given angle until the new angle fits within the desired range. In the case of -800°, we needed to add 360° multiple times until the result was a positive degree measure between 0° and 360°. This was done three times to achieve 280°.
Using this method ensures that any angle can be adjusted to a more standardized measure, making it easier to interpret and work with in mathematical and real-life problems. When approaching these calculations, remember that the essence is bringing your angle to a common framework where comparison and further calculations become straightforward.

Angle Measurement

Angle measurement is the way we quantify the rotation between two intersecting lines or surfaces. Primarily, angles are measured in degrees, where a full circle contains 360 degrees (°).
In the context of coterminal angles, it is vital to understand the complete range of angles, which is from -∞ to ∞ considering real-world scenarios. But practically, we confine ourselves to the range of 0° to 360° for representation and ease.

• Positive angles are measured counterclockwise from the initial side.
• Negative angles are measured clockwise.

For an angle like -800°, the negative sign indicates clockwise movement. The ultimate conversion involves normalizing it to a standard positive angle of 280°. This involves understanding that angles with the same terminal side are inherently equal in their measure of rotation despite potentially having different numerical values.
This concept of angle measurement extends to various applications in trigonometry, physics, engineering, and many more fields, highlighting the importance of correctly understanding and interpreting angles.

Circle Rotations

Circle rotations relate to the complete turn around a circle's circumference, which is equivalent to 360 degrees. This concept is foundational when dealing with angles because adding 360° to an angle results in a rotation that ends up at the same location on the circle.
For the coterminal angle scenario, recognizing that every full rotation (360°) leaves you at the same terminal side allows you to adjust any angle to your desired range. As seen in the solution of the given problem, adding 360° to -800° adjusted the original angle in successive full rotations:

• First rotation: -800° to -440°
• Second rotation: -440° to -80°
• Third rotation: -80° to 280°

The concept helps to understand not only how angles convert into one another but also how angles operate in repeating cycles.
Understanding circle rotations is also pivotal when interpreting repetitive, periodic phenomena in real-world systems such as clocks, waves, and orbits, where the same positions recur systematically with every full cycle.