Problem 14 - Applied Calculus

Question

For each angle: i. Find the radian measure (without using a calculator). ii. Check your answers using a graphing calculator. a. \(750^{\circ}\) b. \(150^{\circ}\) c. \(10^{\circ}\)

Step-by-Step Solution

Answer

750°: \(\frac{25\pi}{6}\), 150°: \(\frac{5\pi}{6}\), 10°: \(\frac{\pi}{18}\).

Step 1: Converting Angles to Radians

To convert degrees to radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). Let's apply this to each angle.

Step 2: Convert 750° to Radians

Using the formula: \( 750^{\circ} \times \frac{\pi}{180} = \frac{750 \pi}{180} \). Simplifying, we have \( \frac{25\pi}{6} \).

Step 3: Convert 150° to Radians

Using the formula: \( 150^{\circ} \times \frac{\pi}{180} = \frac{150 \pi}{180} \). Simplifying, we get \( \frac{5\pi}{6} \).

Step 4: Convert 10° to Radians

Using the formula: \( 10^{\circ} \times \frac{\pi}{180} = \frac{10 \pi}{180} \). Simplifying further, the result is \( \frac{\pi}{18} \).

Step 5: Verification with Graphing Calculator

Verify each of these results by using a graphing calculator to convert degree measures to radians, ensuring the results match the manual calculations: \( \frac{25\pi}{6} \), \( \frac{5\pi}{6} \), and \( \frac{\pi}{18} \).

Key Concepts

Radians Conversion

Radians provide an alternative way to measure angles. Instead of using degrees, one full revolution around a circle is measured as a total of \(2\pi\) radians. To convert from degrees to radians, you can use the formula:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
This formula relies on the fact that \(180^{\circ}\) is equal to \(\pi\) radians, making \(\pi/180\) a useful conversion factor.

• Multiply the number of degrees by \(\pi/180\)
• Simplify the fraction if necessary

Converting angles to radians is a common task in trigonometry and calculus, providing a more natural mathematical approach.
For example, converting \(750^{\circ}\) involves multiplying by \(\pi/180\), resulting in \(\frac{25\pi}{6}\) radians after simplification.

Graphing Calculator Verification

Graphing calculators are powerful tools for math students. They can help verify the accuracy of manual calculations, especially when converting angles from degrees to radians. To verify, input the degree value into the calculator and use the built-in conversion function.

• Enter the degree value (e.g., \(750^{\circ}\))
• Use the radian conversion function or mode
• Check that the calculator's output matches the manual result (e.g., \(\frac{25\pi}{6}\))

This approach offers a quick check, ensuring the conversion is correct and building confidence in working with angle measures. While this tool can confirm results, understanding the conversion process remains crucial for deeper learning.

Degrees to Radians Formula

The degrees to radians formula is essential for converting angular measurements. It simplifies the transition between the two systems and is written as:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Here's how you can use it step-by-step:

• Identify the angle in degrees (e.g., \(150^{\circ}\))
• Apply the conversion factor \(\pi/180\)
• Calculate the product and simplify if needed (e.g., \(\frac{5\pi}{6}\) for \(150^{\circ}\))

This conversion is widely used in calculus and physics, as radian measure is more congruent with the natural properties of trigonometric functions. Unlike degrees, radians directly relate arc length with radius, providing more precise measurements and computations.