Problem 52 - Precalculus: A Concise Course

Question

Determine two coterminal angles (one positive and one negative) for each angle. Give your answers in degrees. (a) \(\theta=-390^{\circ}\) (b) \(\theta=230^{\circ}\)

Step-by-Step Solution

Answer

The coterminal angles for \(\theta=-390^{\circ}\) are \(-30^{\circ}\) (positive) and \(-750^{\circ}\) (negative). For \(\theta=230^{\circ}\), the coterminal angles are \(590^{\circ}\) (positive) and \(-130^{\circ}\) (negative).

Step 1: Coterminal angles for \(\theta=-390^{\circ}\)

To find a positive coterminal angle, we add \(360^{\circ}\) to the given angle \(\theta=-390^{\circ}\). This is done to achieve an angle that falls within the positive rotation. Thus, the positive coterminal for \(\theta=-390^{\circ}\) is \(-390^{\circ}+ 360^{\circ}=-30^{\circ}\). Similarly, to find a negative coterminal angle, an additional \(360^{\circ}\) is subtracted from \(\theta\), which gives us \(-390^{\circ}-360^{\circ}=-750^{\circ}\).

Step 2: Coterminal angles for \(\theta=230^{\circ}\)

For \(\theta=230^{\circ}\) determine the positive coterminal by adding \(360^{\circ}\). Here, we get \(230^{\circ} + 360^{\circ}=590^{\circ}\). The negative coterminal is found by subtracting \(360^{\circ}\) from the given \(\theta\), giving \(230^{\circ}-360^{\circ}=-130^{\circ}\).

Key Concepts

Positive Angle

In the world of trigonometry, angles can be expressed as

• positive
• negative

and can be measured in degrees or radians. A positive angle is one that is measured in the counterclockwise direction from the starting point, often called the initial side, to the terminal side.
When dealing with coterminal angles, if the given initial angle is negative or less than zero, we regularly add a full circle, which is \(360^{\circ}\) in degree measurement, to find a positive version.

For example, if you are given an angle like \(-390^{\circ}\), such as in the exercise above, you add \(360^{\circ}\) to find the smallest positive coterminal angle, which in this case is \(-30^{\circ}\). Meanwhile, for an angle already positive like \(230^{\circ}\), adding \(360^{\circ}\) results in a coterminal angle of \(590^{\circ}\).

This practice helps in simplifying angles and finding equivalent angles that are easier to interpret visually and practically when sketching or calculating using them.

Negative Angle

Just like a positive angle moves counterclockwise, a negative angle rotates in the clockwise direction. These angles are often less intuitive because they move against the standard measurement system.
However, understanding negative angles is crucial in calculations and aids in defining positions in opposite directions on a circle.

When working with coterminal angles, to specify a negative angle, you start by subtracting \(360^{\circ}\) from the given angle. If your angle \(\theta\) begins as positive, subtract as follows: for example, from \(230^{\circ}\), the negative coterminal angle will be \(230^{\circ} - 360^{\circ} = -130^{\circ}\). For a negative start, like the \(-390^{\circ}\) angle given in the exercise, subtracting another full rotation brings us further in the negative direction to \(-750^{\circ}\).

Providing both positive and negative coterminal angles for a given angle allows a fuller understanding of its behavior and accessibility from multiple rotational perspectives.

Angle Measurement

Angle measurement is about determining how large an angle is, primarily defining the rotation's extent from one ray or line to another.
Most commonly, angles can be measured in two ways:

• Degrees
• Radians

Degrees are often used in everyday circles and practical applications, given their ease of division.

A full circle equates to \(360^{\circ}\), making it a friendly system for straightforward calculations and partitioning.
Understanding angle measurements also involves recognizing reference angles, which are always between \(0^{\circ}\) and \(360^{\circ}\). This helps in visualizing angles properly, be it positive or negative, and in performing computations with a consistent base measure.
When given a non-standard angle, whether negative or beyond \(360^{\circ}\), you adjust them to fall within understanding by adding or subtracting full circles (\(360^{\circ}\)) to keep computations simpler and more intuitive.

Degrees

Degrees are the unit for angle measurement that most people are familiar with.
In one full rotation around a point, or a complete circle, there are \(360^{\circ}\). Each degree is \(1/360^{\text{th}}\) of a full rotation.
Degrees provide a practical way to measure angles in various fields, including geometry, physics, and navigation.

It’s important to note that determining coterminal angles often involves working within the space of \(360^{\circ}\). Given any angle, you repeatedly add or subtract \(360^{\circ}\) to achieve a measure that is either in standard position or easier to interpret.
This approach is crucial for translating more abstract or complex angle measurements into ones with practical meaning. Solutions like the one shown in the exercise demonstrate how degrees allow these adjustments, making progressions and alternate angle expressions easy to deduce.