Problem 119 - Calculus Volume 1

Question

For the following exercises, convert each angle in radians to degrees. $$\frac{7 \pi}{6} \mathrm{rad}$$

Step-by-Step Solution

Answer

\( \frac{7\pi}{6} \) radians is 210 degrees.

Step 1: Understand the Conversion Formula

To convert an angle from radians to degrees, use the formula: \[ \text{Degrees} = \text{Radians} \times \left( \frac{180}{\pi} \right) \] This formula arises because \( \pi \) radians is equal to 180 degrees.

Step 2: Identify the Given Radians

In this problem, the angle given in radians is \( \frac{7\pi}{6} \) radians. Our task is to convert this into degrees using the conversion formula from Step 1.

Step 3: Apply the Conversion Formula

Substitute the given radians into the formula: \[ \text{Degrees} = \frac{7\pi}{6} \times \left( \frac{180}{\pi} \right) \] Notice how \( \pi \) in the numerator and denominator will cancel out.

Step 4: Simplify the Expression

After canceling the \( \pi \) terms, you are left with: \[ \text{Degrees} = \frac{7}{6} \times 180 \]

Step 5: Calculate the Degrees

Complete the multiplication: \[ \text{Degrees} = \frac{7}{6} \times 180 = 7 \times 30 = 210 \] Thus, \( \frac{7\pi}{6} \) radians is equivalent to 210 degrees.

Key Concepts

Radians to Degrees

Understanding the conversion between radians and degrees is vital in mathematics, especially when dealing with trigonometry and geometry. Radians and degrees are just two different units for measuring angles. It's important to note that while radians might seem less intuitive at first for some students, they are frequently used in higher mathematics and physics.

Here's a simple breakdown to help you remember the conversion:

• Radians tend to be more prevalent in theoretical mathematics because they relate directly to the properties of circles and trigonometric functions.
• Degrees, on the other hand, are often preferred in everyday applications and initial learning phases because they may seem more straightforward.

By mastering both systems, you become more versatile in mathematics.

Conversion Formula

In mathematics, converting an angle from radians to degrees involves a specific formula: \[ \text{Degrees} = \text{Radians} \times \left( \frac{180}{\pi} \right) \]This formula is fundamental in mathematics and stems from the fact that a complete revolution around a circle is \( 2\pi \) radians, which is equivalent to 360 degrees. Breaking it down:

• The factor \( \frac{180}{\pi} \) enables the conversion by equating the two measurements.
• This factor reflects the equality between \( \pi \) radians and 180 degrees.

Knowing how to apply this formula is crucial for solving problems involving radian to degree conversion. Practice is key to getting comfortable with it.

Angle Measurement

Angle measurement is a fundamental aspect of geometry that describes the magnitude of an angle's rotation around its vertex. It's crucial to understand that:

• An angle can be thought of as the measure of turn from one side of a line segment to another.
• The unit of measurement can be in degrees or radians, depending on the requirement of the task.

When measuring angles, it's often useful to relate them to fractions of a circle. For instance, \( \frac{1}{6} \) of a circle can be expressed as 60 degrees or the equivalent in radians. Understanding this can help you navigate both small and large angle measurements effectively.

Mathematics Education

The journey of learning mathematics is often filled with encounters of new concepts such as angle conversion. It's a process of building a robust foundation for understanding more complex ideas. For angle conversions, consider these learning tips:

• Begin with understanding the fundamental relationship between radians and degrees. Visual aids, like circle diagrams, can help solidify this knowledge.
• Practice repeatedly with various angle measures to become fluent in converting between radians and degrees.
• Apply the concepts to real life, where you'll often find angles such as the resolution of screens (measured in degrees) or the movement of gears (often best understood in radians).

In mathematics education, building a strong grasp of these basic conversion concepts will enhance your ability to tackle more advanced topics involving angles.