Problem 32 - Precalculus

Question

Convert from radians to degrees. $$\frac{7 \pi}{6}$$

Step-by-Step Solution

Answer

The angle \( \frac{7\pi}{6} \) radians equals 210 degrees.

Step 1: Understanding the Conversion Formula

To convert from radians to degrees, we use the conversion formula: \[ ext{degrees} = ext{radians} \times \left(\frac{180}{\pi}\right) \]This formula arises from the fact that \( \pi \) radians is equivalent to 180 degrees.

Step 2: Applying the Conversion Formula

Substitute \( \frac{7\pi}{6} \) into the conversion formula:\[ ext{degrees} = \frac{7\pi}{6} \times \left(\frac{180}{\pi}\right)\]This will cancel out \( \pi \), converting it into degrees.

Step 3: Simplifying the Expression

Calculate the multiplication and simplify:\[ ext{degrees} = \frac{7 \times 180}{6} = \frac{1260}{6} = 210\]So, \( \frac{7\pi}{6} \) radians is equivalent to 210 degrees.

Key Concepts

Angle Measurement

Angle measurement is a fundamental concept in mathematics, particularly in geometry and trigonometry. It helps us specify and quantify the direction of a line or shape. There are two main units used to measure angles: degrees and radians.

• Degrees are one of the most commonly used units, where a full circle is 360 degrees.
• Radians offer a different perspective, where a full circle is 2π radians. This means that 180 degrees is equal to π radians.

Understanding both units and how they compare allows a seamless transition between different mathematical topics, making it easier to perform calculations in various areas such as physics and engineering.

Conversion Formula

The conversion from radians to degrees is an essential skill, especially when dealing with trigonometric functions. The formula to convert radians into degrees is: \[\text{degrees} = \text{radians} \times \left(\frac{180}{\pi}\right) \]This formula allows us to easily translate angle measurements, providing consistency across calculations. Here’s how it works:

• The term \( \frac{180}{\pi} \) acts as a conversion factor that transforms radians to degrees.
• Since π radians equals 180 degrees, multiplying by this factor rescales the angle into degrees.

By applying this conversion, you ensure that your angle measurements are suitable for different contexts, such as when presenting results in real-world scenarios or using calculators that require specific units.

Trigonometry

Trigonometry revolves around the study of triangles, especially right triangles, and the relationships between their angles and sides. Angle measurement, whether in radians or degrees, is crucial here.

• In trigonometry, radians are often preferred. This is because they simplify the understanding of oscillatory phenomena, such as waves.
• For practical applications, like engineering or navigation, degrees are more intuitive to comprehend and visualize.

By mastering the conversion between radians and degrees, you can seamlessly integrate the two. This improves flexibility when solving problems in different fields, ensuring you can communicate findings effectively. Understanding the connection between these angles and their trigonometric functions is key to applying mathematical concepts effectively.