Problem 12 - Algebra and Trigonometry

Question

Find the radian measure of the angle with the given degree measure. $$ 15^{\circ} $$

Step-by-Step Solution

Answer

The angle is \(\frac{1}{12}\pi\) radians.

Step 1: Understanding the Problem

We need to convert an angle from degrees to radians. Specifically, we have an angle of \(15^{\circ}\) and we need to express it in radians.

Step 2: Understanding the Conversion Formula

The conversion from degrees to radians is done using the formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \] This is based on the fact that \(180^{\circ}\) is equivalent to \(\pi\) radians.

Step 3: Substituting the Values into the Formula

Substitute \(15\) for degrees into the conversion formula: \[ \text{radians} = 15 \times \frac{\pi}{180} \]

Step 4: Simplifying the Expression

Perform the multiplication and simplify the fraction \(\frac{15}{180}\). This becomes: \[ \frac{15}{180} = \frac{1}{12} \] Therefore, \[ 15^{\circ} = \frac{1}{12} \pi \text{ radians} \]

Key Concepts

Angle Conversion

When we talk about angle conversion, we are referring to the process of transforming one unit of angle measurement to another. Typically, angles are measured in degrees or radians. Degrees are commonly used in everyday activities, such as in maps and geometry, while radians are often utilized in higher mathematics, like calculus and trigonometry. Understanding angle conversion is essential because it allows you to easily translate between different systems of measurement depending on the context you are working in.

• Degrees: Represented with the symbol °, degrees are a more intuitive measure for many due to their common use.
• Radians: A radian is based on the radius of a circle. It provides a natural way of measuring angles during circular motion calculations.

In this exercise, we start with an angle in degrees and aim to convert it into radians, understanding these two perspectives helps frame our calculations.

Degrees to Radians

When converting degrees to radians, it's helpful to remember that one full circle is equal to 360° or \(2\pi\) radians. This relationship provides the basis for any conversion between degrees and radians. To switch from degrees to radians, you multiply the number of degrees by \( \frac{\pi}{180} \). This factor emerges from the fact that \(180^{\circ}\) is equal to \(\pi\) radians.For example, converting \(15^{\circ}\) to radians involves simply multiplying 15 by \(\frac{\pi}{180}\). This straightforward multiplication allows us to move effortlessly from one unit to another. The conversion is important for various applications, especially when dealing with trigonometric functions or calculating arc lengths.

Conversion Formula

The conversion formula between degrees and radians is fundamental in mathematics. It is expressed as: \[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]This equation shows that to convert a degree measure to radians, you multiply by \(\frac{\pi}{180}\). When applying this to \(15^{\circ}\), you would substitute 15 into the formula, resulting in:\[\text{radians} = 15 \times \frac{\pi}{180}\]Continuing to simplify, you divide 15 by 180, resulting in \(\frac{1}{12}\). Finally, you have:\[15^{\circ} = \frac{1}{12} \pi \, \text{radians}\]This formula is not only useful for conversions but also for understanding rotations and angles in various fields such as engineering and physics. Knowing the conversion formula allows us to bridge the gap between the familiar degree system and the mathematically robust radian system.