Problem 43 - Precalculus

Question

Convert from radians to degrees. Round your answers to the nearest hundredth of a degree. 4.

Step-by-Step Solution

Answer

4 radians is approximately 229.18 degrees.

Step 1: Understand the Conversion Formula

To convert radians to degrees, use the formula \( ext{Degrees} = ext{Radians} imes \left( \frac{180}{\pi} \right) \). This formula is derived from the fact that \( 180 \) degrees is equivalent to \( \pi \) radians.

Step 2: Substitute the Given Value into the Formula

We are given the radian measure as 4. Substitute 4 into the formula: \( ext{Degrees} = 4 \times \left( \frac{180}{\pi} \right) \).

Step 3: Calculate the Degrees

Calculate \( 4 \times \left( \frac{180}{\pi} \right) \). First, calculate \( \frac{180}{\pi} \approx 57.2958 \). Then multiply: \( 4 \times 57.2958 = 229.1832 \).

Step 4: Round the Result

Round the calculated degrees to the nearest hundredth. \( 229.1832 \) rounds to \( 229.18 \) degrees.

Key Concepts

Conversion Formula

When diving into the world of angle measurement, radians and degrees are two commonly used units. The conversion formula is crucial in bridging these two systems. The formula is:

• Degrees = Radians × \(\left( \frac{180}{\pi} \right)\).

This is based on the fact that \(180\) degrees is equivalent to \(\pi\) radians, a fundamental relationship in trigonometry. Essentially, to convert any radian measure into degrees, you multiply by the factor \(\left( \frac{180}{\pi} \right)\). This mathematical operation allows you to express the same angle in a different unit of measure, which is very useful in various applications such as physics, engineering, and computer graphics.

Rounding Decimals

Once you have calculated an angle in degrees from radians, it might include a long decimal that isn't very practical. Rounding decimals becomes essential for clean and precise results.

• In this context, rounding to the nearest hundredth means looking at the third decimal place.
• If the third digit is 5 or higher, round up.
• If it is 4 or lower, round down.

For example, a calculated value of \(229.1832\) degrees rounds to \(229.18\) when rounded to two decimal places. Rounding ensures that numerical results are both user-friendly and accurately reflective of the precision needed.

Mathematical Calculation

Performing accurate mathematical calculations is vital when converting radians to degrees. Let's break down the steps involved:

• First, compute the factor \(\frac{180}{\pi}\), which is approximately \(57.2958\).
• Then, multiply the given radian measure (e.g., \(4\)) by this factor.
• Therefore, the calculation would be \(4 \times 57.2958 = 229.1832\).

This structured approach ensures all components of the equation are addressed properly. Each step builds towards achieving a rounded, accurate result. Such detailed calculations highlight the beauty of mathematics, as well as its precision and logical flow.