Problem 64 - Calculus for Biology and Medicine

Question

(a) Convert \(-15^{\circ}\) to radian measure. (b) Convert \(\frac{3}{4} \pi\) to degree measure.

Step-by-Step Solution

Answer

(a) -\frac{\pi}{12} ; (b) 135°.

Step 1: Understanding Degree to Radian Conversion

The formula to convert degrees to radians is given by \[\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\]where \(\theta_{deg}\) is the angle in degrees and \(\theta_{rad}\) is the angle in radians.

Step 2: Applying the Formula to Convert -15° to Radians

Using the formula from Step 1, substitute \(-15\) for \(\theta_{deg}\):\[\theta_{rad} = -15 \times \frac{\pi}{180} = -\frac{15\pi}{180}\]Simplify the fraction:\[-\frac{15\pi}{180} = -\frac{\pi}{12}\]

Step 3: Understanding Radian to Degree Conversion

To convert radians to degrees, use the formula:\[\theta_{deg} = \theta_{rad} \times \frac{180}{\pi}\]where \(\theta_{rad}\) is the angle in radians.

Step 4: Applying the Formula to Convert \(\frac{3}{4}\pi\) to Degrees

Substitute \(\frac{3}{4}\pi\) for \(\theta_{rad}\) into the formula from Step 3:\[\theta_{deg} = \frac{3}{4}\pi \times \frac{180}{\pi}\]Cancel out \(\pi\) and calculate:\[\theta_{deg} = \frac{3}{4} \times 180 = 135^{\circ}\]

Key Concepts

Degree to Radian Conversion

Converting degrees to radians is a fundamental concept in trigonometry, and it involves using a specific mathematical formula. This formula is particularly useful since many scientific and engineering applications require angle measurements in radians rather than degrees.

• The conversion formula is: \( \theta_{rad} = \theta_{deg} \times \frac{\pi}{180} \), where \( \theta_{deg} \) represents the angle in degrees.
• This formula leverages the fact that \( 180^{\circ} \) is equivalent to \( \pi \) radians, facilitating the conversion between these two units.

When converting a specific angle, such as \(-15^{\circ}\), to radians:1. Substitute the degree value into the formula: \[ \theta_{rad} = -15 \times \frac{\pi}{180} \]2. Simplify the calculation: \[ \theta_{rad} = -\frac{15\pi}{180} = -\frac{\pi}{12} \]For ease in memorizing: always think of it as multiplying the degree by \( \frac{\pi}{180} \).

Radian to Degree Conversion

The conversion from radians to degrees is essentially the reverse process of converting from degrees to radians. Many students find radians more abstract, so transforming them back to degrees can provide a clearer understanding of the angle's measurement.

• To convert from radians to degrees, use the formula: \( \theta_{deg} = \theta_{rad} \times \frac{180}{\pi} \).
• This conversion is essential when you encounter radian measures in exercises or applications and wish to express them in degrees.

For example, convert \( \frac{3}{4} \pi \) radians to degrees:1. Substitute the radian value into the formula: \[ \theta_{deg} = \frac{3}{4} \pi \times \frac{180}{\pi} \]2. Simplify by canceling out \( \pi \) and multiply: \[ \theta_{deg} = \frac{3}{4} \times 180 = 135^{\circ} \]This process will turn any radian measure back into our familiar degree system effortlessly.

Mathematical Formulas

Using mathematical formulas in trigonometric conversions might initially seem complex, but understanding their derivation and application simplifies the process.

• Remember, these conversions hinge upon the principal that \( 180^{\circ} = \pi \) radians, allowing for a seamless transition between units.
• The formula from degrees to radians: \( \theta_{rad} = \theta_{deg} \times \frac{\pi}{180} \).
• Conversely, from radians to degrees: \( \theta_{deg} = \theta_{rad} \times \frac{180}{\pi} \).

These formulas are universally applicable, letting you convert any angle measurement with confidence. By applying these carefully, you'll not only succeed in exercises but also deepen your trigonometric comprehension. Remember to always double-check your simplifications and calculations for potential errors. With practice, these conversions will feel almost automatic.