Problem 13 - Algebra and Trigonometry

Question

\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ \frac{7 \pi}{6} $$

Step-by-Step Solution

Answer

210 degrees

Step 1: Understand Radian to Degree Conversion

To convert radians to degrees, use the formula: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]This formula uses the fact that \( \pi \) radians is equal to 180 degrees.

Step 2: Substitute Given Radian Measure

Substitute the given radian measure, \( \frac{7\pi}{6} \), into the formula:\[ \text{Degrees} = \frac{7\pi}{6} \times \frac{180}{\pi} \]

Step 3: Cancel Out the Pi

In the expression \( \frac{7\pi}{6} \times \frac{180}{\pi} \), the \( \pi \) in the numerator and the denominator cancel each other out. You're left with:\[ \text{Degrees} = \frac{7}{6} \times 180 \]

Step 4: Perform the Multiplication

Now, multiply \( \frac{7}{6} \) by 180:\[ \text{Degrees} = 7 \times 30 = 210 \]

Step 5: Conclusion

The degree measure of the angle given in radian measure \( \frac{7\pi}{6} \) is 210 degrees.

Key Concepts

Angle Measurement

When discussing the measurement of angles, we often utilize two units: degrees and radians. Understanding how angles are measured is crucial for studying geometry, trigonometry, and several fields of science. Degrees are perhaps the most familiar. A full circle is divided into 360 degrees, providing a straightforward way to describe an angle. This method is akin to slicing a pie into 360 even slices.

On the other hand, radians offer a more mathematical approach to angle measurement. In this system, an angle is considered in relation to the radius of a circle. A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius.

• A full circle equals \(2\pi\) radians.
• Half a circle, or 180 degrees, equals \(\pi\) radians.
• Therefore, a right angle is \(\frac{\pi}{2}\) radians, equivalent to 90 degrees.

These two systems allow for flexibility, depending on the situation at hand. Converting between them is essential for problem-solving in mathematics.

Mathematical Conversion

Converting angles from radians to degrees or vice versa requires a solid understanding of the mathematical relationship between these units. The key formula for conversion relies on the equivalence of \(\pi\) radians to 180 degrees.

The conversion formula from radians to degrees is:

• \(\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\)

This formula stems from simple proportional reasoning. Because 180 degrees equals \(\pi\) radians, any radian measure can be converted to degrees by determining how many halves of \(\pi\) fit into it and scaling it by 180 degrees.

For example, converting \(\frac{7\pi}{6}\) radians to degrees involves multiplying by \(\frac{180}{\pi}\). Throughout this conversion, the \(\pi\) terms cancel, simplifying the calculation to multiplying fractions, which students often find more intuitive.

Degrees and Radians

Understanding the relationship between degrees and radians is a fundamental aspect of trigonometry and geometry. Each system serves different purposes and is useful for different types of calculations.

Degrees offer a straightforward approach for describing angles, especially in contexts like navigation or everyday situations. However, radians are invaluable in higher mathematics due to their relation to the unit circle and their seamless integration with calculus.

• A radian provides a direct link to the relationship between a circle's arc length and its radius.
• This unit is particularly useful for measuring periodic functions and derivatives in calculus.

Mastering this duality not only aids in solving geometric problems but also deepens your understanding of math's underlying structure. Switching between these two measurements may seem daunting at first, but with practice, it becomes a natural and fluid process.