Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$-0.57$$

Step-by-Step Solution

Answer

The conversion of -0.57 radians into degrees gives approximately -32.647 degrees.

Step 1: Identify the given angle and conversion factor

The given angle in radians is -0.57 rad. The conversion factor from radians to degrees is \(\frac{180^\circ}{\pi \text{ rad}}\).

Step 2: Multiply the given radian value with the conversion factor

To convert the measure of the angle from radians to degrees, the given radian value should be multiplied by the conversion factor. It is done as follows: -0.57 rad * \(\frac{180^\circ}{\pi \text{ rad}}\).

Step 3: Calculate and round off to three decimal places

After performing the multiplication, the obtained value should be rounded to three decimal places to get the final answer.

Key Concepts

Radian to Degree Conversion

When working with angles, it is important to understand the difference between radians and degrees. The radian is a unit of angular measure based on the radius of a circle. It's the angle created when the arc length equals the radius of the circle. Degrees, on the other hand, divide a circle into 360 equal parts. To convert an angle from radians to degrees, we use the conversion factor \(\frac{180^\circ}{\pi}\). This is because \(\pi\) radians is equivalent to 180 degrees. Here's a quick way to remember the conversion:

• If you have radians and want degrees, multiply by \(\frac{180^\circ}{\pi}\).
• If you have degrees and want radians, multiply by \(\frac{\pi}{180^\circ}\).

For instance, converting \(-0.57\) radians to degrees, you multiply:\[-0.57 \times \frac{180^\circ}{\pi}\]Following this will help ensure you get the correct degree measurement.

Angle Measurement

Understanding how angles are measured helps in interpreting different types of data in mathematics and physics. Angles can be defined in several ways, but they most commonly appear as measurements in radians or degrees.

• Radians are a more natural form of measurement used in calculus and applied mathematics due to their relationship with the unit circle.
• Degrees are most commonly used in day-to-day settings due to their ease of understanding, where a full circle equals 360 degrees.

Every angle can be described in either format, and converting between radians and degrees is crucial when solving various mathematical problems. Let's explore how such measurements apply in practical contexts. Suppose you're working with trigonometric functions. These typically operate using radian measurements, so converting to degrees beforehand can be vital to make sense of the results.

Rounding Decimals

Rounding decimals plays a critical role in math, especially when dealing with irrational numbers or results from calculations. In many exercises, results may have to be rounded to a specific number of decimal places for simplicity or to match precision requirements.Here's how to round decimals to three places:

• Identify the number at the third decimal place.
• Look one digit to the right (fourth decimal place).
• If this digit is 5 or higher, round the third decimal up. If it is lower, keep the third decimal place as is.

For example, if you have the number \(3.14159\), to round to three decimal places, look at the fourth decimal digit (which is 5). According to rounding rules, you increase the third decimal by one, turning \(3.141\) into \(3.142\).Always remember to carefully carry each calculation step to precise decimal points to maintain accuracy, rounding only at the end to ensure an exact result.