Problem 66 - Precalculus : Building Concepts and Connections

Question

Convert each angle from radians to degrees. $$\frac{\pi}{45}$$

Step-by-Step Solution

Answer

The angle \(\frac{\pi}{45}\) radians in degrees is 4 degrees.

Step 1: Identify the value in radians.

We have an angle expressed in radians as \(\frac{\pi}{45}\). This will be the value that will need to be converted to degrees.

Step 2: Use the conversion factor.

To convert from radians to degrees, we multiply by \(\frac{180}{\pi}\). This is because \(\pi\) radians is equivalent to 180 degrees.

Step 3: Perform the multiplication.

Multiply \(\frac{\pi}{45}\) by \(\frac{180}{\pi}\) to get the answer in degrees. The \(\pi\) in the numerator and the \(\pi\) in the denominator will cancel out. Hence, the calculation would be \(\frac{180}{45}\).

Step 4: Simplify the Result

Simplify the fraction \(\frac{180}{45}\) to get the final answer, which is 4 degrees.

Key Concepts

Radian Measure

Understanding the radian measure is essential for converting angles into degrees. A radian is a unit of angle measure that is based on the radius of a circle. In a circle, when you take an arc length equal to the radius of the circle, the angle subtended by that arc at the center of the circle is one radian. Circumference of a circle can be calculated by using the formula
\(2\frac{\times \textrm{\textrm{pi}} \textrm{\times radius}}\right\), which evaluates to \(2\frac{\times \textrm{\textrm{pi}} \textrm{\times radius}}\right\), making the full circle's measure \(2\pi\) radians. Therefore, \(\frac{\textrm{\textrm{pi}}}{45}\) radian, as mentioned in our exercise, is a much smaller portion of the circle.

Degree Measure

Degree measure is another common way to express angles. Unlike radian measure, which is based on the radius of a circle, the degree measure divides one full rotation of a circle into 360 equal parts, known as degrees. This division stems from ancient civilizations and has been standardized for its ease of use in various applications. Therefore, when we talk about converting radians to degrees, we are translating one system of angle measurement into another more commonly used in everyday life and various fields like navigation and geometry.

Conversion Factor

The conversion factor acts as a bridge between radian measure and degree measure. This factor is derived from the relationship between the circumference of a circle and the degrees in a circle. Since the full circle contains \(2\pi\) radians and 360 degrees, the ratio of degrees to radians is \(360:2\pi\), or simply \(180:\pi\). To convert radians to degrees, we multiply the number of radians by this conversion factor, \(\frac{180}{\pi}\), which effectively changes the units from radians to degrees. Understanding and using the conversion factor is pivotal in performing accurate angle conversions.

Angle Conversion

The process of angle conversion from radians to degrees follows a simple multiplication using the conversion factor. Starting with the radian measure \(\frac{\pi}{45}\), we apply the conversion factor by performing the multiplication
\(\frac{\pi}{45} \times \frac{180}{\pi}\).
The \(\pi\) symbols cancel each other, which leaves us with the simplified expression \(\frac{180}{45}\). Upon performing the division, we arrive at the angle in degrees, which, in this example, is 4 degrees. It's essential to simplify the fraction completely to achieve the final degree measure. For more complex angles, the conversion may require additional steps, but the principle remains consistent: multiply the radian value by the conversion factor and simplify.