Problem 63 - Calculus for Biology and Medicine

Question

(a) Convert \(65^{\circ}\) to radian measure. (b) Convert \(\frac{11 \pi}{12}\) to degree measure.

Step-by-Step Solution

Answer

(a) \( \frac{13\pi}{36} \) radians; (b) \(165^{\circ}\).

Step 1: Understanding Degree to Radian Conversion

To convert degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This factor comes from dividing the circle (360 degrees) with radians (2π).

Step 2: Applying the Conversion for 65 Degrees

Multiply \(65^{\circ}\) by the conversion factor \( \frac{\pi}{180} \):\[65 \times \frac{\pi}{180} = \frac{65\pi}{180}\].Reduce \( \frac{65\pi}{180} \) by dividing both numerator and denominator by 5:\[ \frac{65\pi}{180} = \frac{13\pi}{36} \text{ radians} \].

Step 3: Understanding Radian to Degree Conversion

To convert radians to degrees, use the conversion factor \( \frac{180}{\pi} \), which comes from the same relationship between a full circle in degrees and radians.

Step 4: Applying the Conversion for \( \frac{11\pi}{12} \)

Multiply \( \frac{11\pi}{12} \) by the conversion factor \( \frac{180}{\pi} \):\[\frac{11\pi}{12} \times \frac{180}{\pi} = \frac{11 \times 180}{12} = \frac{1980}{12}\].Divide \(1980\) by \(12\) to get the degree measure:\[165^{\circ}\].

Key Concepts

Degree to Radian Conversion

When dealing with angles in mathematics, sometimes you'll need to convert from degrees to radians. This is a common practice, especially in calculus and trigonometry. The conversion factor used here is \( \frac{\pi}{180} \). This arises because a full circle is 360 degrees and is equivalent to \( 2\pi \) radians. So, to convert any angle from degrees to radians, you multiply the degree measure by this conversion factor.

For example, to convert \( 65^{\circ} \) to radians, you multiply:

• \( 65 \times \frac{\pi}{180} \)
• This simplifies to \( \frac{65\pi}{180} \)
• Further simplifying by dividing both numerator and denominator by 5 gives \( \frac{13\pi}{36} \)

Hence, \( 65^{\circ} \) equals \( \frac{13\pi}{36} \) radians. This step-by-step approach ensures you convert correctly and understand the underlying reason behind each step.

Radian to Degree Conversion

Converting radians back to degrees is just as essential as the reverse process. The conversion factor for this is \( \frac{180}{\pi} \). This factor comes from the same equivalence of 360 degrees to \( 2\pi \) radians.

Let's convert \( \frac{11\pi}{12} \) radians into degrees:

• Multiply \( \frac{11\pi}{12} \) by \( \frac{180}{\pi} \)
• This results in \( \frac{11 \times 180}{12} \)
• This value simplifies to \( \frac{1980}{12} \)
• After calculating, you get \( 165^{\circ} \)

So, \( \frac{11\pi}{12} \) radians is equivalent to \( 165^{\circ} \). Understanding this conversion allows you to easily switch between these two forms, which is highly useful in various mathematical contexts.

Mathematical Conversions

Mathematical conversions, like those between degrees and radians, are foundational in understanding and working comfortably with angles. These conversions rely on well-established relationships between different measurement systems, like those used in geometry, calculus, and physics.

Some key concepts to remember include:

• A full circle in degrees is \( 360^{\circ} \) and in radians is \( 2\pi \).
• The conversion factor from degrees to radians is \( \frac{\pi}{180} \).
• Conversely, the conversion factor from radians to degrees is \( \frac{180}{\pi} \).
• Simplifying fractions is crucial to ensure the final answer is in the simplest form possible.

Understanding and applying these conversion techniques allows you to fluidly transition between units, which is an invaluable skill in solving diverse mathematical problems.