Problem 12 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

15° is \(\frac{\pi}{12}\) radians.

Step 1: Understand the Conversion Formula

The formula to convert an angle from degrees to radians is given by \( \text{radians} = \frac{\pi}{180} \times \text{degrees} \). We'll use this formula to convert 15 degrees into radians.

Step 2: Substitute the Given Value into the Formula

Substitute the degree measure, \( 15^{\circ} \), into the conversion formula: \( \text{radians} = \frac{\pi}{180} \times 15 \).

Step 3: Simplify the Expression

Simplify the expression \( \frac{\pi}{180} \times 15 \) by dividing both the numerator and the denominator by their greatest common divisor, which is 15. This gives us \( \frac{\pi}{12} \).

Step 4: Conclusion

Thus, the radian measure of an angle of 15 degrees is \( \frac{\pi}{12} \).

Key Concepts

Degree to Radian Conversion

Converting degrees to radians is a fundamental skill in precalculus mathematics. Understanding the conversion formula is critical for navigating between these two common angle measurements. The radian is the standard unit of angular measure used in many areas of mathematics. Here's how you can convert degrees to radians easily:

• Use the formula: \( \text{radians} = \frac{\pi}{180} \times \text{degrees} \) to convert from degrees to radians.
• The function \( \frac{\pi}{180} \) acts as the conversion factor, because \( 180^{\circ} \) is equivalent to \( \pi \) radians, creating a direct proportion.

For example, to find the radian measure of \( 15^{\circ} \), you substitute it into the formula: \[ \text{radians} = \frac{\pi}{180} \times 15 \].
Simplifying this expression, we use common factors to reduce to \( \frac{\pi}{12} \). This shows that an angle of \( 15^{\circ} \) corresponds to \( \frac{\pi}{12} \) radians.

Angle Measurement

Angles can be measured in degrees or radians, two units that allow us to understand the size of an angle. Each system has its own applications in mathematics and other fields. Here is a simple breakdown of both measurements:

• Degrees: A full circle is divided into 360 equal parts, and each part is a degree.
• Radians: A circle is also expressed in terms of \( 2\pi \) radians, where one full revolution measures \( 2\pi \).

Using radians may be more prevalent in higher mathematics, as it simplifies many mathematical models and formulas. In contexts where angles repeatedly interact with circular functions, radians provide a tighter integration with those calculations, which can make them very beneficial. Understanding how to measure angles in both degrees and radians allows for a more versatile approach to problem-solving in trigonometry and geometry.

Precalculus Mathematics

Precalculus mathematics sets the foundation for calculus and involves crucial skills such as understanding functions, complex numbers, and the properties of geometric figures, including angles. When studying angles in precalculus, students often need to become comfortable switching between degrees and radians. This flexibility enhances understanding in calculus, where the radian measure is often preferred due to its natural fit into the calculus framework.

• Focus areas include trigonometric functions, their graphs, and transformations, which rely heavily on angle measures.
• Additionally, real-world problems and more advanced math sequences commonly require using radians for their efficiencies in calculations.

Mastering these concepts in precalculus ensures that students are well-prepared for the rigors of calculus, as well as for fields like physics and engineering, where these skills are put to practical use.