Problem 23 - Basic Technical Mathematics with Calculus

Question

Use a calculator conversion sequence to change the given angles in radians to equal angles expressed in degrees to the nearest \(0.01^{\circ}\) $$-4.110 \mathrm{rad}$$

Step-by-Step Solution

Answer

The angle \(-4.110\) radians converts to approximately \(-235.48^{\circ}\).

Step 1: Identify the conversion factor

To convert radians to degrees, the conversion factor is \ \(\frac{180}{\pi}\) \ because there are \(\pi\) radians in \(180^{\circ}\).

Step 2: Setup the conversion equation

To convert from radians to degrees, multiply the radian measure by \ \(\frac{180}{\pi}\).\ For the angle \(-4.110\) radians, the equation will be: \ \(-4.110 \times \frac{180}{\pi}\)."

Step 3: Perform the calculation

Use a calculator to compute: \ \(-4.110 \times \frac{180}{\pi} \approx -235.4787\).

Step 4: Round the result

Round the computed degree measure to the nearest \(0.01^{\circ}\). \ This results in \(-235.48^{\circ}\).

Key Concepts

Conversion Factor

When converting radians to degrees, understanding the conversion factor is crucial. The conversion factor between radians and degrees is \( \frac{180}{\pi} \). This relationship exists because a full circle is \( 360^{\circ} \), which corresponds to \( 2\pi \) radians. Therefore, half of this, or \( \pi \) radians, is equivalent to \( 180^{\circ} \). Knowing this, anytime you wish to convert radians into degrees, multiply the radian measure by \( \frac{180}{\pi} \).
\( \pi \) acts as a bridge between the two units. Whether for trigonometry or other fields, this factor efficiently shifts your angle from one unit to the other, ensuring you're working in the correct measurement system. Remember, this conversion is vital for comparing angles in different contexts, such as math and engineering.

Calculator Conversion Sequence

To convert angles from radians to degrees efficiently, using a calculator is often your best bet. Calculators simplify the multiplication process, especially with irrational numbers like \( \pi \). Here's the sequence you follow:

• First, identify the radian value you want to convert. For instance, \(-4.110\) radians.
• Then, input this radian measure into your calculator.
• Multiply it by the conversion factor, \( \frac{180}{\pi} \), using the calculator's multiplication function.
• Press the equal sign to get your result in degrees.
• Lastly, make sure your calculator is set to degree mode if you plan to perform further operations.

Performing these steps allows you to obtain degrees promptly, reducing error compared to manual computation. Calculators handle the complexities of \( \pi \) and ensure precision in the conversion process.

Rounding to Nearest Degree

Once you've performed your conversion, the next step is rounding. Typically, degrees are rounded to the nearest hundredth, which means looking at the second decimal place and adjusting accordingly.

• Calculate the degree value using your calculator. For instance, \(-235.4787^{\circ}\).
• Check the third decimal place (if it exists) to determine how to round the second.
• If the third decimal is 5 or higher, round the second decimal up. If it's 4 or lower, keep it unchanged.
• This results in a degree measure rounded to \(0.01^{\circ}\). For our example, it becomes \(-235.48^{\circ}\).

Rounding is essential for ease of interpretation and comparison, especially in practical applications where excessive precision may not be necessary. It's a small but important step that ensures your results are both accurate and user-friendly.