Problem 52 - Precalculus

Question

For the following exercises, find the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal to the given angle. $$ 700^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is 340°.

Step 1: Understand the Concept

Coterminal angles are angles that share the same terminal side when drawn in standard position. They differ by a whole number of complete rotations, which is equivalent to adding or subtracting multiples of 360 degrees.

Step 2: Calculate the Initial Remainder

To find a coterminal angle between \(0^\circ\) and \(360^\circ\), first subtract \(360^\circ\) from the given angle until the result is between \(0^\circ\) and \(360^\circ\). Start with the given angle, which is \(700^\circ\).

Step 3: Subtract 360 Degrees

Subtract \(360^\circ\) from \(700^\circ\):\[700^\circ - 360^\circ = 340^\circ\].

Step 4: Verify the Remainder

Check if \(340^\circ\) is between \(0^\circ\) and \(360^\circ\). Since it is, \(340^\circ\) is the coterminal angle you are looking for.

Key Concepts

Angle Measurement

Understanding angle measurement is crucial in identifying coterminal angles. Angles are measured in degrees, where a full circle is divided into 360 degrees. This is a common unit in trigonometry, allowing for a consistent way to describe the rotation around a point. Each degree divides the circle into 1/360th of the total circumference. This measurement helps us correlate physical rotations or geometric positions with numerical values. In complex problems, precise angle values allow you to accurately determine relative positions or interactions between angles. When asked to find a specific angle, such as those between 0° and 360°, it is important because it limits the solution set to one full rotation, simplifying calculations and interpretations.

Full Rotations

A full rotation refers to a complete turn around a circle, which corresponds to 360 degrees. Similarly, a half rotation is 180 degrees, and a quarter rotation is 90 degrees. Understanding full rotations is key in finding coterminal angles. When we say that angles are coterminal, it means they have completed full rotations and still share the same terminal position as another angle. For example, an angle of 720 degrees has completed two full rotations because 2 times 360 is 720. When calculating coterminal angles, adding or subtracting multiples of a full rotation (360°) determines how many complete cycles an angle has undergone. This concept is especially helpful to visualize and solve angle problems that extend beyond one full revolution around the circle.

Standard Position

The standard position of an angle is a classical concept in geometry and trigonometry. It is defined when:

• The vertex is located at the origin of a coordinate system.
• The initial side lies along the positive x-axis.
• This standard setup helps ensure that angle measurements are consistent for interpretation and analysis.

In standard position, the angle's direction of rotation is also crucial. Counterclockwise rotation indicates positive angles, while clockwise represents negative angles. This standardization simplifies the understanding and calculation of angle-related problems, as it provides a uniform reference. Knowing the standard position also allows one to explicitly state where the terminal side of an angle lands. Therefore, this structured approach is particularly useful when dealing with coterminal angles, as it serves as a clear start point for considering angle equivalences through rotations.

Subtracting Multiples

Subtracting multiples involves manipulating angle measurements to find equivalents within a specified range, typically between 0 degrees and 360 degrees. This is essential for determining coterminal angles. When given an angle like 700 degrees, the first step is to understand it may need adjustment to fit within a single circle rotation. By subtracting 360 until the angle lies within the 0° to 360° range, you effectively strip away the completed full rotations: For example:

• 700° - 360° gives 340°

The angle 340° is coterminal with 700° because they land on the same terminal side after counting full rotations. This method of subtracting is a practical computational step, ensuring that angles are easily comparable and fit standard conventions of measurement.