Problem 15 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with each given angle. $$ -356^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle for \(-356^{\circ}\) between \(0^{\circ}\) and \(360^{\circ}\) is \(4^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side. Therefore, an angle is coterminal with another angle if the difference between them is a multiple of \(360^{\circ}\). In other words, if an angle \(A\) is coterminal with an angle \(B\), then \(A = B + 360n\), where \(n\) is an integer.

Step 2: Converting the given negative angle to positive

In this case, we have an angle of \(-356^{\circ}\). To find its coterminal angle in the range from \(0^{\circ}\) to \(360^{\circ}\), we add multiples of \(360^{\circ}\) until we get an angle in the desired range. We only need to add \(360^{\circ}\) once, as \(-356^{\circ} + 360^{\circ} = 4^{\circ}\), which is in the interval from \(0^{\circ}\) to \(360^{\circ}\). Therefore, the coterminal angle for \(-356^{\circ}\) within the range of \(0^{\circ}\) to \(360^{\circ}\) is \(4^{\circ}\).

Key Concepts

Angle Measurement

Understanding angles is crucial for solving problems related to coterminal angles. Angles can be measured in degrees, which are symbols that determine the amount of rotation from the starting position. The circle is divided into 360 equal parts, known as degrees. When we discuss angles ranging from \(0^{\circ}\) to \(360^{\circ}\), we refer to the full rotation of a circle. A positive angle involves a counter-clockwise rotation from the positive x-axis, whereas a negative angle indicates a clockwise rotation.
Recognizing where the angle sits between \(0^{\circ}\) and \(360^{\circ}\) helps you understand which part of the rotation the angle represents. For instance, an angle like \(-356^{\circ}\) means the position is almost a full clockwise rotation, just 4 degrees short of a complete turn.

Angle Conversion

Converting angles to find their coterminal counterparts is a fundamental exercise in trigonometry. When given a negative angle, like \(-356^{\circ}\), we want to find an equivalent positive angle within a standard circle rotation (between \(0^{\circ}\) and \(360^{\circ}\)).

To convert, simply add \(360^{\circ}\) to your given angle. This happens because a full circle amounts to a complete \(360^{\circ}\). Adding \(360^{\circ}\) to \(-356^{\circ}\), we get \(4^{\circ}\), which is the coterminal angle.

• This conversion shows that manipulating angles within their rotational congruence leads to equivalents that are easier to grasp visually.
• Use these principles to toggle between negative and positive angles effortlessly.

Conversions are about repositioning angles into a friendlier format for analysis.

Integer Multiples

A cornerstone in understanding coterminal angles is the concept of integer multiples. By adding or subtracting full rotations of \(360^{\circ}\) (which can be expressed as \(360n\)), where \(n\) is any integer, we locate coterminal angles.

This integer manipulation is essential because:

• Integer multiples help identify angles that have the same terminal side despite different rotations.
• For \(-356^{\circ}\), we added \(360 \times 1\) to normalize it into the positive range, resulting in \(4^{\circ}\).

Understanding this process reveals how angles can be translated into different forms without altering their geometric position. Use integer multiples to seamlessly transition angles for calculating problems related to rotation and periodicity.