Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Convert each radian measure to degrees. $$\frac{2 \pi}{3}$$

Step-by-Step Solution

Answer

The radian measure \( \frac{2\pi}{3} \) is equal to 120 degrees.

Step 1: Understand the Conversion Formula

To convert a radian measure to degrees, we use the conversion formula: \[ 1 ext{ radian} = \frac{180}{\pi} ext{ degrees} \] This means that to convert radians to degrees, we multiply the radian measure by \( \frac{180}{\pi} \).

Step 2: Apply the Conversion Formula

Using the conversion formula, we'll convert \( \frac{2 \pi}{3} \) radians to degrees. The formula to apply is: \[ \frac{2 \pi}{3} \times \frac{180}{\pi} \] This will give us the angle in degrees.

Step 3: Simplify the Expression

Cancel out the \( \pi \) in the numerator and the denominator: \[ \frac{2 \cdot 180}{3} \] This simplifies to: \[ \frac{360}{3} \]

Step 4: Perform the Division

Divide 360 by 3 to find the degree measure: \[ 360 \div 3 = 120 \] Thus, the radian measure \( \frac{2 \pi}{3} \) converts to 120 degrees.

Key Concepts

Angle Conversion

Converting angles is an essential skill in both mathematics and physics. Nicely understanding how to do this can make math concepts clearer. When converting, you should remember that angles can be expressed in different units, the most common being degrees and radians. Each unit has its own unique advantages; hence the need for conversion arises in different contexts.

• Degrees: Degrees are the more conventional unit. They are commonly used in everyday applications. A full circle is 360 degrees.
• Radians: Radians offer a more natural way for mathematicians to tackle trigonometric calculations. In this system, a full circle is expressed as \( 2\pi \) radians.

Knowing how to convert between these two units enables flexibility in calculations. The conversion formula is a staple tool here. Simply by multiplying a radian measure by the conversion factor \( \frac{180}{\pi} \), you obtain the angle in degrees. This factor is derived from the fact that \( \pi \) radians equal 180 degrees.

Radian Measure

The radian measure comes from the mathematics of circles. It might feel less intuitive than degrees at first, but it's very useful. A radian measures the angle created when we take the radius of a circle and wrap it along the circle's edge. In simpler terms, it's the angle at the center of a circle that describes an arc equal to the radius length.

• Relation to circle circumference: Since the circumference of a circle is \( 2\pi\) times the radius, a full circle has an angle of \( 2\pi\) radians.
• Radian usage: Radians are especially handy in calculus and trigonometry, as they simplify certain mathematical properties and formulas.

In our example, \( \frac{2\pi}{3} \) radians, we're looking at a portion of the circle that is a fraction less than one full \( \pi \) radian. Converting this measure to degrees can sometimes make it easier to visualize for practical purposes.

Degrees Conversion

Converting radians into degrees is straightforward once you know the conversion factor. In general, to get degrees from radians, you use the factor \( \frac{180}{\pi} \). This relationship is based on matching radians to a circle's geometry.To convert, we multiply the radian measure by \( \frac{180}{\pi} \), essentially translating everyday angles into a more mathematical form:

• For \( \frac{2\pi}{3} \) radians: the expression becomes \( \frac{2\pi}{3} \times \frac{180}{\pi} \).
• This simplifies to \( \frac{2 \times 180}{3} \) since \( \pi\)'s cancel out.
• The division \( \frac{360}{3} \) gives us 120, resulting in 120 degrees.

This method of conversion is systematic and once familiar, can quickly be applied to any radian measure, turning it into an angle understandable in the everyday language of degrees.