Problem 23 - Algebra and Trigonometry

Question

In Exercises \(21-28,\) convert each angle in radians to degrees. $$ \frac{2 \pi}{3} $$

Step-by-Step Solution

Answer

The angle of \(\frac{2 \pi}{3}\) radians is equivalent to 120 degrees.

Step 1: Identify the Radian Measure

First, identify the measure given in radians, which is \(\frac{2 \pi}{3}\) in this case.

Step 2: Convert Radians to Degrees

To convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\), which is the conversion factor. Therefore, you'll compute the value as: \(\frac{2 \pi}{3}\) * \(\frac{180}{\pi}\). The \(\pi\) in the numerator and denominator will cancel out, leaving \(\frac{2}{3}\) * 180.

Step 3: Final Calculation

Finally, perform the multiplication: \(\frac{2}{3}\) * 180 = 120.

Key Concepts

Radian Measure

In mathematics, angles can be measured in different units, with radians being a fundamental one alongside degrees. A radian is a measure based on the radius of a circle. One complete revolution around a circle is equal to an angle of \(2\pi\) radians. This means that if you walk around a circle's circumference from one point all the way back to the starting point, the angle turned is \(2\pi\) radians.
Here's how it works in simpler terms:

• The total radians in a circle is \(2\pi\).
• Half of a circle translates to \(\pi\) radians, or 180 degrees.

Understanding radian measure gave mathematicians a tool to express angles in relation to the geometry of a circle, making calculations more natural for circles and periodic functions.Radian measure is particularly useful in trigonometry and calculus because it simplifies many mathematical expressions, especially when compared to degrees.

Angle Conversion

Converting from radians to degrees is an essential skill in both academic studies and practical applications. Since one full circle (\(2\pi\) radians) is equivalent to 360 degrees, we can set up a simple conversion factor based on this relationship.Here's how the conversion works:

• Start with your radian measure (e.g., \(\frac{2\pi}{3}\)).
• Multiply by the conversion factor \(\frac{180}{\pi}\).
• Cancel out the \(\pi\) in both the numerator and the denominator.

This conversion factor comes from the equivalence \(\pi\) radians equals 180 degrees. By multiplying \(\frac{2\pi}{3} \) by \(\frac{180}{\pi}\), you effectively swap radians for degrees.
Through this simple process, you convert radians, which are inherently tied to a circle's radius, into a more intuitive measurement (degrees), which is often easier to visualize and use in common applications.

Mathematical Calculation

The final part of the conversion process involves performing a straightforward mathematical calculation. Once the radian measure is multiplied by the conversion factor, the equation is simplified by canceling out any common terms. Here's a detailed look:

• After identifying that \(\frac{2\pi}{3}\) is your starting point, multiply it by \(\frac{180}{\pi}\).
• Cancelling out \(\pi\) from numerator and denominator results in \(\frac{2}{3} \cdot 180\).
• The multiplication \(\frac{2}{3} \times 180\) gives you 120.

This means that the angle \(\frac{2\pi}{3} \) radians is equivalent to 120 degrees.
Breaking down the calculation into simpler steps ensures accuracy and makes the process more digestible. Practicing these steps can help students build confidence and ensure they can apply this mathematical calculation reliably in various contexts.