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Chapter 1: Problem 45

For each given angle name the quadrant in which the terminal side lies. $$ -\frac{13 \pi}{8} $$

Short Answer

Expert verified
The terminal side of \(-\frac{13\text{蟺}}{8} \) lies in the first quadrant.

Step by step solution

01

- Convert the angle to degrees

To understand the location of the terminal side, first convert the angle from radians to degrees. Use the conversion factor: \( 1 \text{ radian} = \frac{180}{\text{蟺}} \text{ degrees} \).So, the conversion for the given angle \( -\frac{13\text{蟺}}{8} \) is:\[ -\frac{13\text{蟺}}{8} \times \frac{180}{\text{蟺}} = -\frac{13 \times 180}{8} = -292.5^\text{掳} \]
02

- Determine the Equivalent Angle

Since angles are periodic and the terminal side of angles coterminal with a given angle lies in the same quadrant, find a positive coterminal angle by adding or subtracting full revolutions (\(360^\text{掳}\)).\[ -292.5^\text{掳} + 360^\text{掳} = 67.5^\text{掳} \]
03

- Identify the Quadrant

The angle \(67.5^\text{掳}\) lies in the first quadrant, since it's greater than \(0^\text{掳}\) and less than \(90^\text{掳}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle Conversion
Understanding how to convert angles is crucial in trigonometry. Angles can be measured in degrees or radians. A radian is a standard unit of angular measure, used in many areas of mathematics. To convert an angle from radians to degrees, we use the relationship:
  • \( 1 \text{ radian} = \frac{180}{\text{蟺}} \text{ degrees} \)
To convert the given angle, \( -\frac{13\text{蟺}}{8} \text{ radians} \), to degrees, we multiply by \( \frac{180}{\text{蟺}} \). Starting with the given angle, the conversion formula becomes:
  • \( -\frac{13\text{蟺}}{8} \times \frac{180}{\text{蟺}} = -292.5^\text{掳} \)
By using this formula, you can easily convert any angle from radians to degrees. It's essential to understand both units of measure as they are commonly used in different contexts.
Coterminal Angles
Coterminal angles are angles that share the same terminal side when drawn in standard position. This concept is important because it helps simplify problems by reducing angles to a common reference. To find a positive coterminal angle, we can add or subtract multiples of a full revolution, which is \(360^\text{掳}\). For the given angle, \( -292.5^\text{掳} \), we can find a coterminal angle by adding \(360^\text{掳}\):
  • \( -292.5^\text{掳} + 360^\text{掳} = 67.5^\text{掳} \)
Additionally, you could find other coterminal angles by continuing to add or subtract \(360^\text{掳}\):
  • Add: \(67.5^\text{掳} + 360^\text{掳} = 427.5^\text{掳} \)
  • Subtract: \(-292.5^\text{掳} - 360^\text{掳} = -652.5^\text{掳} \)
By understanding how to find coterminal angles, you can often simplify problems involving angle measurements and make them more manageable.
Quadrant Identification
Identifying the quadrant in which an angle's terminal side lies is a fundamental skill in trigonometry. The coordinate plane is divided into four quadrants:
  • First Quadrant: \(0^\text{掳}\) to \(90^\text{掳}\)
  • Second Quadrant: \(90^\text{掳}\) to \(180^\text{掳}\)
  • Third Quadrant: \(180^\text{掳}\) to \(270^\text{掳}\)
  • Fourth Quadrant: \(270^\text{掳}\) to \(360^\text{掳}\)
After converting an angle to degrees and finding any necessary coterminal angles, you can determine its quadrant. For the given angle, \(-\frac{13\text{蟺}}{8} \text{ radians}\) which converts to \(67.5^\text{掳}\), we see it lies within \(0^\text{掳}\) to \(90^\text{掳}\). Therefore, the terminal side of the angle is in the first quadrant. Understanding quadrant identification helps in solving problems involving trigonometric functions and their signs in different quadrants.

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