Problem 15 - Precalculus

Question

Convert from degrees to radians. Leave the answers in terms of \(\pi\). $$45^{\circ}$$

Step-by-Step Solution

Answer

45 degrees is equal to \( \frac{\pi}{4} \) radians.

Step 1: Understanding the Conversion Factor

To convert from degrees to radians, we use the conversion factor that relates degrees to radians. Specifically, we know that 180 degrees is equivalent to \( \pi \) radians.

Step 2: Setting Up the Conversion

To convert 45 degrees to radians, we use the ratio \( \frac{\pi}{180} \) to convert degrees to radians. We will multiply 45 degrees by \( \frac{\pi}{180} \).

Step 3: Simplifying the Expression

Perform the multiplication: \( 45 \times \frac{\pi}{180} \). Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 45.

Step 4: Calculating the Result

\( \frac{45}{180} = \frac{1}{4} \). Thus, when multiplied by \( \pi \), the expression becomes \( \frac{\pi}{4} \).

Key Concepts

Radian

The term 'radian' is fundamental in trigonometry and geometry. A radian is a unit of angular measure used in many areas of mathematics. Simply put, it is the angle created when the radius of a circle is wrapped along its edge. In more technical terms, a radian is the angle subtended at the center of the circle by an arc with the same length as the circle's radius.
If you imagine a circle with a radius of 1, the circumference of the entire circle would be approximately 6.283 radians (since the circumference of a circle is given by the formula \(2\pi\)).
This means that there are \(2\pi\) radians in a full circle, just like there are \(360^{\circ}\) in a full circle.

• 1 radian is approximately 57.2958 degrees.
• It's a dimensionless value which makes it very useful in calculus.

The use of radians simplifies many mathematical calculations, which is why it is preferred in calculus and higher-level mathematics.

Degree

A degree is a more familiar unit of measurement for angles, especially in everyday contexts. We often use degrees to describe angles in geometry or any practical fields requiring precision.
A degree is defined as \(\frac{1}{360}\) of a circle's rotation. This division makes full-circle rotations easily comparable and understandable in everyday situations, such as in navigation and construction.

• There are \(360^{\circ}\) in a full circle.
• Degrees are more intuitive for most people because of the 360 division.

Understanding how degrees measure angles can make it easier for people to visualize and process angular measurements in both theoretical and practical applications.

Conversion Factor

The conversion factor between degrees and radians is crucial for converting one measurement to the other. Understanding this will help anyone easily switch between radians and degrees.
The main conversion factor is \(\pi\) radians equals \(180^{\circ}\). This means you can convert degrees to radians by using this formula:

• To convert from degrees to radians: multiply the degree value by \(\frac{\pi}{180}\).
• To convert from radians to degrees: multiply the radian value by \(\frac{180}{\pi}\).

For example, if we convert \(45^{\circ}\) to radians, we multiply it by \(\frac{\pi}{180}\), leading to a result of \(\frac{\pi}{4}\).
Remembering this conversion factor is essential in connecting and comparing the two angular units. It gives us a straightforward method for switching back and forth between degrees and radians as needed in various calculations and real-world applications.