Problem 25 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{11 \pi}{4}\)

Step-by-Step Solution

Answer

\(\frac{11\pi}{4}\) in degrees is \(495^\circ\).

Step 1: Convert Radian to Degrees

To convert from radians to degrees, use the formula \( heta( ext{degrees}) = heta( ext{radians}) \times \frac{180}{ ext{π}} \). This formula arises because \( ext{π} ext{ radians} = 180^ ext{o} \).

Step 2: Substitute the Given Radian

Substitute \( \frac{11 ext{π}}{4} \) into the conversion formula to find its degree measure: \( \frac{11 ext{π}}{4} \times \frac{180}{ ext{π}} \).

Step 3: Simplify the Expression

Notice that \( \text{π} \) in the numerator and denominator cancels out, simplifying the expression to \( \frac{11}{4} \times 180 \).

Step 4: Compute the Multiplication

Calculate \( \frac{11}{4} \times 180 \), which results in \( \frac{1980}{4} \).

Step 5: Perform Division to Get the Answer

Divide 1980 by 4 to get the degree measure: \( 495^ ext{o} \).

Key Concepts

Radian Measure

A radian is a unit of angular measure used in many areas of mathematics. It is based on the radius of a circle. Imagine wrapping a length of the circle's radius around its edge; this length would define an angle of one radian at the circle's center. This makes radians a natural way to express angles when dealing with circular or periodic objects.

Unlike degrees, which divide a circle into 360 equal parts, radians divide the circumference of a circle in relation to its radius. Typically, a full circle covers an angle of \(2\pi\) radians. Thus, radians provide a straightforward way to relate angles to physical dimensions.

Degree Measure

Degrees offer a familiar way to measure angles in everyday settings. A degree is defined as \(\frac{1}{360}\) of a complete rotation around a circle's center.

Degrees are easy to visualize and thus frequently employed in fields like geography and engineering. However, when dealing with functions of periodic nature or differential equations, radians become more convenient.

• One full circle: 360 degrees
• One right angle: 90 degrees
• Degrees breakdown allows easy measurement and communication

Mathematical Formula

Transforming between radians and degrees is essential for problems involving angular measurements. This is where a mathematical formula becomes invaluable:

Conversion from radians to degrees utilizes the relationship: \( \theta(\text{degrees}) = \theta(\text{radians}) \times \frac{180}{\pi} \).
Here's the reasoning: Since \(\pi\) radians equal 180 degrees, multiplying a radian measure by \(\frac{180}{\pi}\) converts it to degrees.

• It translates radians (a more abstract concept) into degrees (a more tangible one).
• Provides an effortless way to switch between the two measurement units.

Simplification Steps

Converting \( \frac{11\pi}{4} \) radians to degrees involves a series of simplification steps:

• **Multiply by the conversion factor.** Substitute the given radian measure into the conversion formula, multiplying by \(\frac{180}{\pi}\).
• **Cancel out values.** Notice that \(\pi\) itself cancels out, leaving you with a simpler expression.
• **Perform arithmetic calculations.** You are left with the operation \(\frac{11}{4} \times 180\).
• **Finalize with division.** After performing the multiplication, divide the result \(1980\) by \(4\) to obtain the final answer: \(495^\circ\).

These steps allow you to transition from a mathematical expression in radians to a more practical degree measure. In many educational and practical scenarios, such clear understanding of these basic transformations is crucial.