Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Convert the following degree measures to radians (leave \(\pi\) in your answer). (a) \(30^{\circ}\) (b) \(45^{\circ}\) \(370^{\circ}(\mathrm{c})\) (c) \(-60^{\circ}\) (d) \(240^{\circ}\) (e) \(-370^{\circ}\) (f) \(10^{\circ}\)

Step-by-Step Solution

Answer

(a) \(\frac{\pi}{6}\), (b) \(\frac{\pi}{4}\), (c) \(\frac{37\pi}{18}\), (d) \(-\frac{\pi}{3}\), (e) \(\frac{4\pi}{3}\), (f) \(-\frac{37\pi}{18}\), (g) \(\frac{\pi}{18}\).

Step 1: Understanding Degree to Radian Conversion

To convert an angle from degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This means that you multiply the degree measure by \( \frac{\pi}{180} \) to find the equivalent radian measure.

Step 2: Convert 30° to Radians

Multiply 30 by \( \frac{\pi}{180} \):\[ 30^{\circ} \times \frac{\pi}{180} = \frac{30\pi}{180} = \frac{\pi}{6} \]Therefore, \( 30^{\circ} = \frac{\pi}{6} \) radians.

Step 3: Convert 45° to Radians

Multiply 45 by \( \frac{\pi}{180} \):\[ 45^{\circ} \times \frac{\pi}{180} = \frac{45\pi}{180} = \frac{\pi}{4} \]Thus, \( 45^{\circ} = \frac{\pi}{4} \) radians.

Step 4: Convert 370° to Radians

Multiply 370 by \( \frac{\pi}{180} \):\[ 370^{\circ} \times \frac{\pi}{180} = \frac{370\pi}{180} = \frac{37\pi}{18} \]Thus, \( 370^{\circ} = \frac{37\pi}{18} \) radians.

Step 5: Convert -60° to Radians

Multiply -60 by \( \frac{\pi}{180} \):\[ -60^{\circ} \times \frac{\pi}{180} = \frac{-60\pi}{180} = -\frac{\pi}{3} \]Therefore, \( -60^{\circ} = -\frac{\pi}{3} \) radians.

Step 6: Convert 240° to Radians

Multiply 240 by \( \frac{\pi}{180} \):\[ 240^{\circ} \times \frac{\pi}{180} = \frac{240\pi}{180} = \frac{4\pi}{3} \]Therefore, \( 240^{\circ} = \frac{4\pi}{3} \) radians.

Step 7: Convert -370° to Radians

Multiply -370 by \( \frac{\pi}{180} \):\[ -370^{\circ} \times \frac{\pi}{180} = \frac{-370\pi}{180} = -\frac{37\pi}{18} \]Thus, \( -370^{\circ} = -\frac{37\pi}{18} \) radians.

Step 8: Convert 10° to Radians

Multiply 10 by \( \frac{\pi}{180} \):\[ 10^{\circ} \times \frac{\pi}{180} = \frac{10\pi}{180} = \frac{\pi}{18} \]Thus, \( 10^{\circ} = \frac{\pi}{18} \) radians.

Key Concepts

Angle Conversion

Converting angles from degrees to radians is a fundamental skill in trigonometry. Degrees and radians are two units for measuring angles, and knowing how to switch between them is essential. The key element here is the conversion factor, which is \( \frac{\pi}{180} \). This arises from the fact that a full circle is \( 360^{\circ} \) or \( 2\pi \) radians. Therefore:

• \( 180^{\circ} = \pi \) radians
• To convert degrees to radians, multiply by \( \frac{\pi}{180} \)
• To convert radians to degrees, multiply by \( \frac{180}{\pi} \)

This simple conversion factor allows you to navigate between these two systems quickly. So if you have an angle of \( 45^{\circ} \), you multiply it by \( \frac{\pi}{180} \) to get \( \frac{\pi}{4} \) radians. Angle conversion is not only a mathematical exercise; it is crucial for applications in physics, engineering, and computer graphics.

Trigonometry

Trigonometry is the branch of mathematics that deals with the study of triangles, particularly right-angled triangles. It involves the relationships between the angles and lengths of triangles. Trigonometry uses the concept of the unit circle to explore these relationships further, where angles can be in degrees or radians.

• In the unit circle, the circle's radius is 1.
• Radians are a natural choice for measuring angles here because they relate directly to the arc length. For instance, an angle of \( \frac{\pi}{2} \) radians corresponds to a quarter arc of the circle.
• Basic trigonometric functions (sine, cosine, and tangent) are often easier to work with in radians because of their periodic properties.

Understanding how to convert angles between degrees and radians is critical for applying these trigonometric functions effectively. It also aids in solving real-world problems such as wave motion and oscillations.

Mathematics Education

Mathematics education aims to build a solid foundation for understanding key concepts such as angle conversion. Teaching methods focus on making these topics accessible and engaging for learners. Effective mathematics education involves:

• Using practical applications and visual aids to help students grasp abstract concepts.
• Providing step-by-step examples, much like those seen in homework exercises, to allow students to follow along and understand the reasoning behind each step.
• Encouraging students to practice frequently with a variety of exercises ensures deeper comprehension and skill mastery.

Teachers and educational platforms must illustrate how concepts like degrees and radians are used in different fields, enhancing students' appreciation of the subject's relevance. Recognizing these connections ensures that learners not only succeed academically but are also prepared for real-world applications.