Problem 27 - Basic Technical Mathematics with Calculus

Question

Use a calculator conversion sequence to change the given angles to equal angles expressed in radians to three significant digits. $$384.8^{\circ}$$

Step-by-Step Solution

Answer

384.8° is approximately 6.72 radians.

Step 1: Understand the Unit Circle

Angles can be measured in degrees or radians. A complete circle is 360 degrees, which is the same as \(2\pi\) radians. Therefore, one degree equals \(\frac{\pi}{180}\) radians.

Step 2: Set Up the Conversion Formula

To convert degrees to radians, use the formula: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \).

Step 3: Insert the Given Degree Value

Substitute the angle in degrees to the conversion formula: \( 384.8 \times \frac{\pi}{180} \).

Step 4: Perform the Calculation

Calculate the product using a calculator: \( 384.8 \times \frac{3.14159}{180} \), which results in approximately \( 6.71555 \).

Step 5: Round to Three Significant Digits

Round the result to three significant digits to get the final answer: 6.72. Thus, \( 384.8^{\circ} \) is approximately \( 6.72 \) radians when rounded to three significant figures.

Key Concepts

Unit Circle

The unit circle is a handy tool for understanding angles and their measurements. Imagine a circle with a radius of 1 unit centered at the origin of a coordinate plane. This circle helps visualize and convert angles between degrees and radians. In the context of the unit circle, because the full circle represents a complete revolution, it equates to 360 degrees or \(2\pi\) radians. Each point on the circle corresponds to an angle measured from the positive x-axis.

• Key Concept: The circumference of the unit circle is \(2\pi\).
• Connection to Angles: The conversion between degrees and radians revolves around this perception.

Understanding this concept helps in various applications, such as trigonometry, physics, and even coding graphics.

Degrees to Radians

Converting degrees to radians is crucial for many mathematical calculations, especially in trigonometry and calculus. To convert an angle from degrees to radians, the formula used is:

• \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)

This formula comes from the relation that \(180^{\circ} = \pi\) radians. By multiplying the degree measure by \(\frac{\pi}{180}\), you change the angle's measurement from degrees to radians.
Let's see an example by converting \(384.8^{\circ}\):

• Insert the degree measure into the formula: \( 384.8 \times \frac{3.14159}{180} \).
• Solve to find the radians, which approximately equals \(6.71555\).

This method can be practiced with any degree measure to find its equivalent in radians.

Significant Figures

Significant figures are a way to express precision in numerical values. In mathematics and science, particularly during conversions and measurements, rounding the result to a specified number of significant figures ensures consistent accuracy. A significant figure shows the precision of a number, indicating which digits are reliable and meaningful.
In our example of converting \(384.8^{\circ}\) to radians:

• The calculated result was approximately \(6.71555\).
• Rounding to three significant figures gives \(6.72\) radians.

Key points to remember about significant figures include:

• The first non-zero digit is always the first significant figure.
• Zeroes between significant figures are also significant.
• Trailing zeros in a decimal are significant.

Carefully considering significant figures provides an accurate representation of your results in calculations.

Calculator Usage

Calculators are essential tools in mathematics, especially for tasks like converting degrees to radians. By correctly using a calculator, you avoid errors and ensure accuracy. Here’s a step-guide on how to use your calculator effectively:

• Start by understanding the functions on your calculator. Ensure it’s set to handle calculations in radians or degrees, depending on your task.
• For conversions, input the degree measure and multiply by \(\frac{\pi}{180}\) using your calculator.
• Make use of the \(\pi\) key for more accurate calculations rather than using the approximation \(3.14159\).
• Use the rounding function, if available, to easily convert the result to the required significant figures, like three significant digits in our example.

Using a calculator efficiently can significantly speed up calculations and improve accuracy, especially in more complex problems.