Problem 21 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(60^{\circ}\)

Step-by-Step Solution

Answer

\(60^{\circ}\) converts to \(\frac{\pi}{3}\) radians.

Step 1: Understanding the Conversion Formula

To convert degrees to radians, you use the formula: \[ ext{Radians} = ext{Degrees} imes \frac{ ext{π}}{180}\]This formula is derived from the fact that \(180^{\circ}\) is equivalent to \(\pi\) radians.

Step 2: Substitute the Degree Value

Substitute the given degree measure into the conversion formula. Here, the degree measure is \(60^{\circ}\): \[ ext{Radians} = 60 \times \frac{\pi}{180}\]

Step 3: Simplify the Expression

Perform the division and simplify the expression to find the radian measure:\[ ext{Radians} = 60 \times \frac{\pi}{180} = \frac{60}{180} \times \pi = \frac{1}{3} \times \pi = \frac{\pi}{3}\]

Step 4: Conclusion

The conversion is complete. \(60^{\circ}\) is equivalent to \(\frac{\pi}{3}\) radians.

Key Concepts

Radian Measure

Radian measure is a way of expressing angles that is particularly useful in mathematics, especially in calculus and trigonometry. Unlike degree measure, which splits a circle into 360 parts, radian measure considers the circle's radius. One radian is the angle made when the arc length of a circle is equal to its radius. This way, a complete circle is naturally 2π radians since the circumference of a circle is 2π times its radius. This concept simplifies many mathematical calculations, as it directly relates the angle to the circle's radius, preserving the natural relationship in circular motion. Think about using radians as speaking the language of circles!

Degree Measure

Degree measure is the more familiar way of expressing angles in day-to-day scenarios. Defined as 1/360 of a complete rotation, degrees are practical because they relate well to the way we divide time and navigation. For example, when you learn basic geometry or use a compass, you will most likely use degrees. So, if you think of a circle having 360 degrees, it intuitively ties to how we dissect daily phenomena—like an hour on a clock (an hour equals 30 degrees). This ancient division of a circle into 360 parts makes it easy to visualize angular sections, especially when relating to real-world scenarios like cutting a pizza or measuring a steep incline.

Conversion Formula

To switch from degrees to radians or vice versa, a conversion formula is essential. The core formula used for converting degrees to radians is:

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

This is based on the equivalence that 180 degrees is equal to π radians. The formula allows for straightforward conversion:

• For example, to convert 60 degrees to radians, multiply 60 by \( \frac{\pi}{180} \), resulting in \( \frac{\pi}{3} \).

Informally speaking, the conversion provides a cheat-sheet for flipping between the two systems efficiently. Remember, π serves as a pivotal reference point, just as 360 degrees offers comparability for whole rotations.

Simplification of Expressions

Simplification of expressions is a key part of converting angular measurements. When dealing with fractions, breaking them down to their simplest form not only makes equations easier to handle but also facilitates deeper understanding. Take the conversion of 60 degrees to radians:

• Start with the basic computation \(60 \times \frac{\pi}{180} \).
• Upon simplification, you divide 60 by 180: \( \frac{60}{180} = \frac{1}{3} \).
• Thus, the simplified answer becomes \( \frac{\pi}{3} \)

These steps exemplify how simplifying expressions helps in producing cleaner, more interpretable results. It’s a crucial skill not only for conversions but for many areas in math where clarity and simplicity lead to easier error-checking and step-saving in computations.