Problem 2 - Calculus with Concepts in Calculus

Question

Convert the following radian measures to degrees. a. \(\frac{\pi}{8}\) b. \(-\frac{3 \pi}{10}\) c. \(\frac{13 \pi}{6}\)

Step-by-Step Solution

Answer

a. 22.5°, b. -54°, c. 390°.

Step 1: Understanding the Conversion Formula

To convert radians to degrees, we use the conversion formula: \[ ext{Degrees} = ext{Radians} imes \frac{180}{\pi} \] This formula is derived from the fact that \( \pi \) radians is equal to 180 degrees.

Step 1: Convert \(\frac{\pi}{8}\) to Degrees

Apply the conversion formula to \( \frac{\pi}{8} \): \[ \text{Degrees} = \frac{\pi}{8} \times \frac{180}{\pi} \] The \( \pi \) cancels out, simplifying to: \[ \text{Degrees} = \frac{180}{8} = 22.5 \] So, \( \frac{\pi}{8} \) radians is 22.5 degrees.

Step 2: Convert \(-\frac{3\pi}{10}\) to Degrees

Apply the conversion formula to \(-\frac{3\pi}{10}\): \[ \text{Degrees} = -\frac{3\pi}{10} \times \frac{180}{\pi} \] The \( \pi \) cancels out, simplifying to: \[ \text{Degrees} = -\frac{3 \times 180}{10} = -54 \] So, \(-\frac{3\pi}{10}\) radians is -54 degrees.

Step 3: Convert \(\frac{13\pi}{6}\) to Degrees

Apply the conversion formula to \(\frac{13\pi}{6}\): \[ \text{Degrees} = \frac{13\pi}{6} \times \frac{180}{\pi} \] The \( \pi \) cancels out, simplifying to: \[ \text{Degrees} = \frac{13 \times 180}{6} = 390 \] So, \(\frac{13\pi}{6}\) radians is 390 degrees.

Key Concepts

Conversion Formula

Converting radians to degrees is an essential skill in trigonometry and geometry. Understanding and applying the conversion formula allows us to switch between these two units of angle measurement with ease. The formula to transform radian measures into degrees is straightforward:

• Radian value multiplied by \( \frac{180}{\pi} \) equals the degree value.

This formula hinges on the relationship between radians and degrees, as \( \pi \) radians is equivalent to 180 degrees. Therefore, the factor \( \frac{180}{\pi} \) is used in the conversion process, enabling us to express any radian measure in terms of degrees by multiplying. For example, when converting \( \frac{\pi}{8} \) radians to degrees, we multiply by \( \frac{180}{\pi} \), and the \( \pi \) in the numerator and denominator cancel out, simplifying our calculation.

Angle Measurement

Angles can be measured in several units, but the two most common are degrees and radians.

• Degrees: A complete circle is divided into 360 degrees. Each degree can be further divided into 60 minutes, and each minute into 60 seconds. This method is historically rooted in ancient civilizations and makes division of the circle into fractions easy.
• Radians: More mathematically pure, radians are based directly on the properties of the circle. One radian is the angle formed when the radius is wrapped along the circle's edge, making the circumference effortlessly related to angle measures.

Understanding both units is vital as they are used differently across various fields. Science and engineering often prefer radians because they simplify many formulas, particularly in calculus and trigonometry. In contrast, everyday scenarios or navigation might use degrees for simplicity and convention.

Trigonometry

Trigonometry is a field of mathematics focused on the relationships between angles and sides of triangles. A large part of trigonometry involves understanding how angle measures can influence these relationships.
Radians and degrees are both crucial in this field but serve different purposes. In trigonometry:

• Many of the functions and identities, such as sine, cosine, and tangent, are traditionally defined with radians, simplifying the derivative processes.
• Converting angles to degrees, however, can make certain applications, like angle construction and measurement in geometry, more intuitive.

Having a good grasp on converting between radians and degrees enhances our ability to solve trigonometric problems effectively. Moreover, it ensures that we apply the correct angle measure in appropriate contexts, whether calculating in a math textbook problem or designing a mechanical system.