Question

The following angles are given in degrees. Convert them to radians: \(30^{\circ}, 45^{\circ}, 90^{\circ}, 180^{\circ}\).

Short answer

- 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

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:\[ \, ext{radians} = \, ext{degrees} \times \left(\frac{\pi}{180}\right) \]

Converting 30° to Radians

Using the formula, we convert 30 degrees to radians:\[ 30^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{30\pi}{180} = \frac{\pi}{6} \, ext{radians} \]

Converting 45° to Radians

Using the conversion formula for 45 degrees:\[ 45^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{45\pi}{180} = \frac{\pi}{4} \, ext{radians} \]

Converting 90° to Radians

For converting 90 degrees, apply the formula:\[ 90^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{90\pi}{180} = \frac{\pi}{2} \, ext{radians} \]

Converting 180° to Radians

Finally, converting 180 degrees to radians gives:\[ 180^{\circ} \times \left(\frac{\pi}{180}\right) = \pi \, ext{radians} \]

Key concepts

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:

• 180 degrees equals \( \pi \) radians.

This relationship forms the basis for conversion between the two units. To convert from degrees to radians, use the formula:

• \( \text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right) \)

The factor \( \frac{\pi}{180} \) arises because 180 degrees correspond to \( \pi \) radians. This fraction scales down the degree measurement to its equivalent in radians, ensuring that arithmetic with radians remains consistent within the realm of circle-related calculations.

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:

• 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.

When dealing with these functions, radians offer a direct route to understanding the periodic nature and properties of waves and oscillations in various scientific fields.

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:

• \( \pi \) 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.

Understanding constants like \( \pi \) provides clarity in mathematical inquiries and ensures accurate problem-solving across multiple domains.