Problem 45 - Precalculus

Question

Convert from radians to degrees. Round your answers to the nearest hundredth of a degree. $$0.85$$

Step-by-Step Solution

Answer

0.85 radians is approximately 48.76 degrees.

Step 1: Understand the Conversion Factor

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

Step 2: Set Up the Conversion

We want to convert \( 0.85 \) radians to degrees. Therefore, we multiply \( 0.85 \) by the conversion factor \( \frac{180}{\pi} \). This can be expressed as: \[ 0.85 \times \frac{180}{\pi} \].

Step 3: Perform the Multiplication

Calculate the value of \( \frac{180}{\pi} \) and then multiply it by \( 0.85 \). The exact calculation is: \[ 0.85 \times 57.2958 \approx 48.756 \].

Step 4: Round to the Nearest Hundredth

Round the answer from the previous step to the nearest hundredth. \( 48.756 \) rounded to the nearest hundredth is \( 48.76 \).

Key Concepts

Understanding Conversion Factors

When converting measurements, a key component is the "conversion factor." This is a numerical factor used to multiply or divide a quantity when converting from one unit of measurement to another. When it comes to converting radians to degrees, the conversion factor is essential. Since there are 180 degrees in \(\pi\) radians, the conversion factor becomes \(\frac{180}{\pi}\).\(\)This tells us how many degrees are in one radian. By multiplying the radian measure by \(\frac{180}{\pi}\),we effectively change the unit from radians to degrees without altering the magnitude of the angle.
It’s like using a "language translator" for numbers, ensuring we maintain the same angle measurement in a different unit. Knowing how to apply this conversion factor is useful in many fields like engineering, physics, and even computer graphics.

The Importance of Rounding Numbers

Rounding is a mathematical technique used to reduce numbers to a simpler, more understandable form. In many scenarios, especially when dealing with endless decimal places, rounding numbers can be very helpful. It provides a way to simplify complex numbers without getting too far from the original value.
When converting from radians to degrees, or in any calculation, rounding usually follows these rules:

• Look at the digit immediately after your rounding digit. If it’s 5 or greater, round up.
• If it’s less than 5, leave the rounding digit as is, disregarding any subsequent digits.

In our example, we rounded the number 48.756 to the nearest hundredth to get 48.76. The hundredth position had the digit 5, which told us to increase the corresponding hundredth place digit from 5 to 6.

Basic Mathematical Operations in Conversion

Mathematical operations form the groundwork of converting units. Performing these operations accurately is crucial in achieving correct conversion results. For converting radians to degrees, we mainly use multiplication.
First, you calculate the conversion factor which involves division \((\frac{180}{\pi}\)). Then, we multiply this factor with the radian value (in our case, 0.85).
Steps to perform mathematical operations in this context include:

• Calculate the exact value of the conversion factor using the constant \(\pi\)\, which can often be approximated as 3.14159.
• Multiply the conversion factor by the radian measure to achieve the measure in degrees.
• Apply rounding as necessary to refine the result into a simpler form, like to the nearest hundredth.

Practicing these basic operations helps in honing skills in handling conversions and applying them in practical scenarios.