Problem 43 - Algebra and Trigonometry

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 is \(280^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side on the coordinate plane. Given an angle, a coterminal angle can be found by adding or subtracting multiples of 360°.

Step 2: Initial Calculation

To find an angle coterminal with \(-800^{\circ}\) between \(0^{\circ}\) and \(360^{\circ}\), start by adding \(360^{\circ}\) to \(-800^{\circ}\) until the result lies within the desired range.

Step 3: First Addition

First, add \(360^{\circ}\) to \(-800^{\circ}\): \(-800^{\circ} + 360^{\circ} = -440^{\circ}\). This angle is still not within the range of \(0^{\circ}\) to \(360^{\circ}\).

Step 4: Second Addition

Add another \(360^{\circ}\) to \(-440^{\circ}\): \(-440^{\circ} + 360^{\circ} = -80^{\circ}\). This angle is also outside the desired range.

Step 5: Third Addition (Finding Coterminal Angle)

Add \(360^{\circ}\) one more time to \(-80^{\circ}\): \(-80^{\circ} + 360^{\circ} = 280^{\circ}\). Now the result is within the range of \(0^{\circ}\) to \(360^{\circ}\).

Step 6: Verifying the Result

Check the calculation to ensure that \(280^{\circ}\) is correct by verifying that \(\ldots, -800^{\circ}, -440^{\circ}, -80^{\circ}, 280^{\circ} \ldots\) are coterminal by means of adding \(360^{\circ}\) multiples.

Key Concepts

Angle Measurement

Understanding angles is vital in geometry and trigonometry. Angles are measured in degrees, which represent a portion of a circle. An entire circle is divided into 360 equal parts, each part being one degree:

• Angles are typically measured from a starting point called the initial side to a terminal side.
• The position of the terminal side determines the measure of the angle.

When working with angles, it's important to note which direction the measurement is in. If you move counterclockwise, the angle is positive. Conversely, moving clockwise results in a negative angle. Many problems require finding angles within a particular range, such as between 0° and 360°.

Coordinate Plane

The coordinate plane is a crucial concept for understanding angles. It's a two-dimensional space made up of two intersecting lines: the x-axis (horizontal) and y-axis (vertical).

• Angles are measured from a point at the origin (0,0), which is where the x-axis and y-axis intersect.
• As you draw an angle, it begins on the initial side, usually lying along the positive x-axis, and rotates to meet the terminal side.

Understanding this plane helps in visualizing angles, especially when determining if two angles are coterminal. Seeing where the terminal sides of angles end up in relation to each quadrant of the plane is essential.

360 Degree

The concept of 360° is fundamental when talking about angles and circles. A full turn around a circle is equivalent to 360°, making it a complete circle. As such:

• Adding or subtracting 360° from an angle results in an angle that looks identical when drawn on the coordinate plane since it's a full rotation.
• This is why we use multiples of 360° to find coterminal angles.

Coterminal angles are separated by whole rotations of 360°, confirming they end at the same position in a circle. This property is applied repeatedly to ensure an angle falls within desired ranges, like between 0° and 360° as highlighted in the exercise.

Negative Angles

Negative angles might seem confusing at first, but they offer distinct insights into rotational directions. When an angle is negative, it indicates clockwise rotation from the initial side:

• Negative angles situate themselves by moving in the opposite direction to positive angles.
• To find a positive coterminal angle from a negative one, continuously add 360° until the result is between 0° and 360°.

For instance, the exercise demonstrates this process by converting -800° into a positive angle that lies within the first full circle of 0° to 360°, ultimately arriving at 280°. This way, any negative angle can be expressed as a positive angle that is coterminal.