Problem 54 - Precalculus

Question

Convert from degrees to radians. Round your answers to three significant digits. $$172^{\circ}$$

Step-by-Step Solution

Answer

172° is approximately 3.00 radians.

Step 1: Identify the Conversion Factor

To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180^{\circ}} \). This means that \( 1^{\circ} = \frac{\pi}{180} \text{ radians} \).

Step 2: Set Up the Conversion Equation

Use the conversion factor to set up the equation for conversion: \( 172^{\circ} \times \frac{\pi}{180^{\circ}} \).

Step 3: Calculate the Radians

Multiply the degree measure by the conversion factor: \( 172 \times \frac{\pi}{180} \). This simplifies to \( \frac{172\pi}{180} \), which can be calculated further.

Step 4: Simplify the Fraction

The fraction \( \frac{172\pi}{180} \) can be slightly simplified based on common divisors. Divide both the numerator and the denominator by 4 to get \( \frac{43\pi}{45} \).

Step 5: Evaluate Numerically

Calculate the value of \( \frac{43\pi}{45} \) using a calculator. Substitute \( \pi \approx 3.14159265358979 \). So, \( \frac{43 \times 3.14159265358979}{45} \approx 3.003 \).

Step 6: Round the Answer

Round the numerical result to three significant figures to obtain the final answer: approximately 3.00 radians.

Key Concepts

Radian Measure

In mathematics, angles can be measured in different ways, and radian measure is one of the fundamental units for this purpose. Unlike degrees, which divide a circle into 360 parts, radians provide a direct link between the diameter of a circle and its circumference. This is achieved by defining a radian as the angle subtended at the center of a circle by an arc equal in length to the circle's radius. This way, a full circle encompasses \(2\pi\) radians. Understanding radians is essential because it allows mathematical formulas involving trigonometry and calculus to become more natural and easier to work with. For example, when dealing with trigonometric functions, their derivatives become simpler when angles are expressed in radians rather than degrees.

Conversion Factor

To convert an angle from degrees to radians, a specific conversion factor is employed. This factor is derived from the relationship between the lengths of arcs and the corresponding angles. The conversion factor is \( \frac{\pi}{180} \). This tells us that one degree equals \( \frac{\pi}{180} \) radians.Using this conversion factor, we multiply the degree value by \( \frac{\pi}{180} \) to obtain the equivalent measure in radians. For example, to convert 172 degrees to radians, we set up the equation as follows:

• 172° × \( \frac{\pi}{180} \)

This conversion not only facilitates the transition from degrees to radians but it also helps in formulating trigonometric expressions, because radians often simplify the mathematical expressions and computations involved.

Simplifying Fractions

When converting degrees to radians, the result is often expressed as a fraction of \( \pi \). Sometimes, these fractions can be simplified to make calculations easier and clearer.Consider the example with 172 degrees. After setting up the conversion equation, we have:

• \( \frac{172\pi}{180} \)

To simplify this fraction, we look for common divisors of the numerator and the denominator. In this case, both 172 and 180 are divisible by 4. So, we divide both by 4:

• \( \frac{172 \div 4}{180 \div 4} = \frac{43\pi}{45} \)

Simplifying fractions not only makes numbers more manageable but also ensures accuracy when dealing with various mathematical computations.

Significant Figures Rounding

In many calculations, particularly in scientific contexts, precision is key. However, there are situations where it is practical to round numbers. This is where significant figures come into play.The concept of significant figures allows us to round the calculated value of a fraction to a desired level of precision. For example, when you evaluate \( \frac{43\pi}{45} \) using a calculator:

• \( \frac{43 \times 3.14159265358979}{45} \approx 3.00342242 \)

To round this to three significant figures, we consider the first three digits:

• Approx. 3.00 radians

Significant figures rounding provides a way to represent numbers concisely, while maintaining an appropriate level of accuracy for the given task. This technique is particularly useful in scientific calculations, where it's essential to balance precision with practicality.