Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Convert the following radian measures to degrees. (a) \(\frac{7}{6} \pi\) (b) \(\frac{3}{4} \pi\) (c) \(-\frac{1}{3} \pi\) (d) \(\frac{4}{3} \pi\) (e) \(-\frac{35}{18} \pi\) (f) \(\frac{3}{18} \pi\)

Step-by-Step Solution

Answer

(a) 210°; (b) 135°; (c) -60°; (d) 240°; (e) -350°; (f) 30°.

Step 1: Understand the Conversion Formula

To convert an angle from radians to degrees, you use the conversion formula: \( 1 \, \text{radian} = \frac{180}{\pi} \, \text{degrees} \). This formula helps transform radian measures into degrees by multiplying the radian measure by \( \frac{180}{\pi} \).

Step 2: Convert (a) \( \frac{7}{6} \pi \) to degrees

First, apply the conversion formula to \( \frac{7}{6} \pi \): \( \frac{7}{6} \pi \times \frac{180}{\pi} = \frac{7 \times 180}{6} \). Simplifying this gives \( 210 \) degrees.

Step 3: Convert (b) \( \frac{3}{4} \pi \) to degrees

Apply the same formula: \( \frac{3}{4} \pi \times \frac{180}{\pi} = \frac{3 \times 180}{4} \). On simplifying, you find the result to be \( 135 \) degrees.

Step 4: Convert (c) \( -\frac{1}{3} \pi \) to degrees

Again, apply the conversion formula: \( -\frac{1}{3} \pi \times \frac{180}{\pi} = -\frac{180}{3} \). This simplifies to \( -60 \) degrees.

Step 5: Convert (d) \( \frac{4}{3} \pi \) to degrees

Multiply \( \frac{4}{3} \pi \) with the conversion factor: \( \frac{4}{3} \pi \times \frac{180}{\pi} = \frac{4 \times 180}{3} \). Simplifying gives \( 240 \) degrees.

Step 6: Convert (e) \( -\frac{35}{18} \pi \) to degrees

Use the formula: \( -\frac{35}{18} \pi \times \frac{180}{\pi} = -\frac{35 \times 180}{18} \). This simplifies to \( -350 \) degrees.

Step 7: Convert (f) \( \frac{3}{18} \pi \) to degrees

Apply the conversion method: \( \frac{3}{18} \pi \times \frac{180}{\pi} = \frac{3 \times 180}{18} \). The final calculation simplifies to \( 30 \) degrees.

Key Concepts

Radian Measure

Radians are an essential unit of angular measurement in mathematics and physics. Unlike degrees, which divide a circle into 360 equal parts, radians divide it based on the circle's radius and arc length. This makes radians particularly useful for calculations involving circles and periodic functions.
One full rotation or circle, which is 360 degrees, is equal to \( 2\pi \) radians. Consequently, \( \pi \) radians represent half a circle, which is 180 degrees.

• Radian measure is dimensionless since it is defined as the ratio of two lengths: the arc length to the radius.
• Common radian values: \( \frac{\pi}{2} = 90 \) degrees, \( \pi = 180 \) degrees, and \( \frac{3\pi}{2} = 270 \) degrees.

Understanding how radians relate to a circle's circumference provides a foundation for converting between radians and degrees.

Degree Measure

Degrees are a widely recognized unit for measuring angles. Originating from ancient astronomers, the degree system divides a circle into 360 equal parts, translating naturally to time and navigation applications.
This segmentation choice is speculated to arise from the Babylonians, who used a base-60 number system, though it conveniently aligns well with a year-long calendar round.

• Degrees are simple to understand and visually intuitive, making them especially popular in fields like navigation and art.
• The degree system can be subdivided into minutes and seconds for more precision, where one degree equals 60 minutes.

Although radians are used frequently in mathematics, degrees remain indispensable for everyday applications such as construction and surveying.

Conversion Formula

Converting between radians and degrees is straightforward with the use of a simple formula:
If you want to convert radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \). Conversely, to convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \).

This formula is fundamental in trigonometry and calculus, where it allows you to switch between units effortlessly.

• Remember that \( 1 \text{ radian} = \frac{180}{\pi} \) degrees.
• It helps to recall that \( \pi \) is approximately 3.14159, simplifying mental calculations.
• Applying the conversion formula requires careful attention to simplify fractions and avoid rounding errors.

With consistent practice, you will find it becomes second nature to apply these conversions in various mathematical contexts.

Mathematics Education

Math education lays the foundation for future learning and application in various disciplines, from engineering to computer science. Understanding concepts like radian and degree measures right from elementary grades aids in building strong analytical and problem-solving skills.
Integrating practical applications, such as using radians in periodic functions or degrees in geometry, can enhance student engagement.

• Incorporating real-world examples helps contextualize these concepts, making them more relatable and less abstract.
• Utilizing visual aids, like circle diagrams, can help students see the relationship between radians, degrees, and the full rotation of a circle.

Well-rounded math education encourages curiosity, nurtures intuitive understanding, and prepares students for complex problem-solving opportunities in higher education and career pathways.