Chapter 8: Problem 1
The following angles are given in degrees. Convert them to radians: \(30^{\circ}, 45^{\circ}, 90^{\circ}, 180^{\circ}\).
Short Answer
Expert verified
- 30° is \( \frac{\pi}{6} \) radians.
- 45° is \( \frac{\pi}{4} \) radians.
- 90° is \( \frac{\pi}{2} \) radians.
- 180° is \( \pi \) radians.
Step by step solution
01
Understanding the Relationship
To convert degrees to radians, we use the relationship that \[ 180^{\circ} = \pi \, ext{radians} \]This allows us to create a conversion factor. The formula to convert degrees to radians is then:\[ \, ext{radians} = \, ext{degrees} \times \left(\frac{\pi}{180}\right) \]
02
Converting 30° to Radians
Using the formula, we convert 30 degrees to radians:\[ 30^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{30\pi}{180} = \frac{\pi}{6} \, ext{radians} \]
03
Converting 45° to Radians
Using the conversion formula for 45 degrees:\[ 45^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{45\pi}{180} = \frac{\pi}{4} \, ext{radians} \]
04
Converting 90° to Radians
For converting 90 degrees, apply the formula:\[ 90^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{90\pi}{180} = \frac{\pi}{2} \, ext{radians} \]
05
Converting 180° to Radians
Finally, converting 180 degrees to radians gives:\[ 180^{\circ} \times \left(\frac{\pi}{180}\right) = \pi \, ext{radians} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Angle Conversion
Angles can be expressed in different units, and in mathematics, we frequently convert between degrees and radians. Each type of unit has its applications and benefits. Degrees represent a full circle as 360 degrees, making it intuitive for beginners. Radians, on the other hand, are fundamental in higher mathematics, particularly in calculus and trigonometry.
In the angle conversion process, it is essential to remember the key relationship:
Understanding how to convert between these measurements fluently enhances your ability to work with various geometric and trigonometric problems.
In the angle conversion process, it is essential to remember the key relationship:
- 180 degrees equals \( \pi \) radians.
- \( \text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right) \)
Understanding how to convert between these measurements fluently enhances your ability to work with various geometric and trigonometric problems.
Basics of Trigonometry
Trigonometry revolves around the study of triangles, especially right-angled ones, and it plays a significant role in mathematics and physics. By understanding the relationships between the angle and lengths in a triangle, trigonometry provides tools to solve complex geometry problems.
In trigonometry:
Developing fluency with degrees and radians helps explore these concepts thoroughly, providing a foundation to delve into more intricate trigonometric problems and calculus-based applications.
In trigonometry:
- Angles can be measured using degrees or radians.
- Degrees are straightforward for practical measurements, while radians offer mathematical simplicity in calculations.
- Trigonometric functions such as sine, cosine, and tangent express these relationships and are often defined in terms of radians.
Developing fluency with degrees and radians helps explore these concepts thoroughly, providing a foundation to delve into more intricate trigonometric problems and calculus-based applications.
Role of Mathematical Constants
Mathematical constants are essential figures in mathematics, representing values that define some of the fundamental principles and formulas used across various branches. Among these, \( \pi \) is one of the most recognized constants, vital for calculations involving circles.
\( \pi \) is approximately equal to 3.14159 and represents the ratio of a circle's circumference to its diameter. Its significance extends beyond simple geometry, becoming a foundational constant used in calculus, trigonometry, and engineering.
When performing angle conversions:
\( \pi \) is approximately equal to 3.14159 and represents the ratio of a circle's circumference to its diameter. Its significance extends beyond simple geometry, becoming a foundational constant used in calculus, trigonometry, and engineering.
When performing angle conversions:
- \( \pi \) aids in translating angular measures from degrees to radians, as one-half of the full circle in radians is represented as \( \pi \).
- This mathematical constant bridges geometric intuition in degrees with analytical ease found in radians.