Problem 1 - Calculus with Concepts in Calculus

Question

Convert the following degree measures to radians. a. \(210^{\circ}\) b. \(-405^{\circ}\) c. \(1^{\circ}\)

Step-by-Step Solution

Answer

a. \( \frac{7\pi}{6} \) radians; b. \(-\frac{9\pi}{4}\) radians; c. \(\frac{\pi}{180}\) radians.

Step 1: Understanding the Conversion Formula

To convert degrees to radians, we use the conversion formula: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \). This formula stems from the fact that \(180^{\circ} \) is equivalent to \( \pi \) radians.

Step 2: Convert 210° to Radians

Apply the conversion formula to \(210^{\circ}\):\[210^{\circ} \times \frac{\pi}{180} = 210 \times \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6}.\]Hence, \(210^{\circ}\) is equal to \( \frac{7\pi}{6} \) radians.

Step 3: Convert -405° to Radians

Apply the conversion formula to \(-405^{\circ}\):\[-405^{\circ} \times \frac{\pi}{180} = -405 \times \frac{\pi}{180} = \frac{-405\pi}{180} = -\frac{9\pi}{4}.\]Thus, \(-405^{\circ}\) is equal to \(-\frac{9\pi}{4} \) radians.

Step 4: Convert 1° to Radians

Use the conversion formula for \(1^{\circ}\):\[1^{\circ} \times \frac{\pi}{180} = \frac{\pi}{180}.\]Therefore, \(1^{\circ}\) is equal to \( \frac{\pi}{180} \) radians.

Key Concepts

Radians

Radians are an alternative measure of angles used in mathematics, especially in trigonometry and calculus. Instead of measuring angles in degrees—which is more common in everyday usage radians provide a more natural way to work with angles in mathematical calculations.

In a circle, radians can be understood by considering the circumference. A full circle is equal to 360 degrees, which is also equal to \(2\pi\) radians. Therefore, one radian represents the angle created when you take the radius and wrap it around the circle's edge.

• 1 full circle = 360 degrees = \(2\pi\) radians
• 1 radian = \(\frac{180}{\pi}\) degrees
• 1 degree = \(\frac{\pi}{180}\) radians

By using radians, calculations for various functions, such as sine and cosine in trigonometry, become more straightforward. This efficiency arises because radians relate directly to the radius and circumference, making them essential for mathematical analysis and problem-solving.

Conversion Formula

The conversion formula is your tool for changing degrees into radians, which is a vital skill in trigonometry and other math-related fields. The formula itself is simple and based on the equivalence between degrees and radians.

To convert degrees to radians, use the formula:

• \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)
• This means if you multiply the number of degrees by \(\frac{\pi}{180}\), you'll get the measure in radians.
• The conversion factor \(\frac{\pi}{180}\) arises from the relation \(180^{\circ} = \pi \) radians.

This formula is not just a math trick; it's built upon the geometry of a circle. Every time you perform this conversion, you're taking advantage of the natural link between linear and circular measurements. Practicing this conversion will help solidify your understanding of how degrees and radians interact in real mathematical problems.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between angles and lengths in right-angled triangles. However, its concepts extend far beyond, playing a crucial role in calculus, physics, engineering, and even computer graphics.

Radians serve as a significant unit in trigonometry because of their close relationship to the unit circle. In trigonometry:

• The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system.
• Radians provide a way to measure angles in this circle that neatly corresponds to arc lengths.

Use of radians simplifies the application of trigonometric functions like sine, cosine, and tangent. These functions often use radians as inputs because they correspond directly to the coordinates on the unit circle. Understanding radians and how to convert from degrees is essential. It ensures that when you calculate these functions, you get meaningful and mathematically sound results.