Problem 23 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(-225^{\circ}\)

Step-by-Step Solution

Answer

The degree measure \(-225^{\circ}\) is equal to \(-\frac{5\pi}{4}\) radians.

Step 1: Understand the conversion formula

To convert degrees to radians, use the formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]

Step 2: Substitute the degree measure

Insert the given degree measure into the formula. \[ -225^{\circ} \times \frac{\pi}{180} \]

Step 3: Simplify the expression

Simplify the fraction: \[ -225 \div 180 = -\frac{5}{4} \] So the expression becomes: \[ -\frac{5}{4} \times \pi = -\frac{5\pi}{4} \]

Step 4: Conclusion

The degree measure \(-225^{\circ}\) is equivalent to \(-\frac{5\pi}{4}\) radians.

Key Concepts

Degrees to Radians

Degrees and radians are two common units for measuring angles. Converting between them can be crucial, especially in fields like mathematics or physics. Unlike degrees, which divide a full circle into 360 parts, radians are based on the radius of a circle. A complete circle in radians is equal to \(2\pi\). To convert degrees into radians, we need to understand their relationship with circles and fractions of \(\pi\). Learning this conversion is not only useful for performing calculations but also helps deepen your understanding of angles and their representations.

Conversion Formula

The key to converting from degrees to radians lies in the conversion formula. The formula is simple and straightforward:

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

This formula basically tells us how many radians correspond to one degree. Let’s say you have an angle in degrees and you want to convert it to radians. You multiply the degree value by \(\frac{\pi}{180}\). The fraction \(\frac{\pi}{180}\) serves as a conversion factor that changes degrees into radians. It is derived from the fact that a full circle is \(360^{\circ}\) or \(2\pi\) radians, which simplifies to \(\pi = 180^{\circ}\). This makes the conversion process accurate and reliable.

Simplification Process

Simplification is an important step when converting angles, as it can make calculations more manageable. After substituting the degrees into the conversion formula, you may end up with a fraction that needs simplification. For example, with \(-225^{\circ}\):

• First, plug the degree value into the formula: \(-225^{\circ} \times \frac{\pi}{180}\)
• Next, simplify the numbers \(-225\) and \(180\). Divide both by their greatest common divisor. Here, \(-225\) divided by \(180\) gives \(-\frac{5}{4}\).

So the simplified conversion to radians is \(-\frac{5\pi}{4}\). Simplification is crucial because it makes your answer more precise and easier to interpret in subsequent calculations.

Negative Angles

Negative angles might seem tricky, but they are simply angles measured in the opposite direction (clockwise) on a circle. When converting negative angles from degrees to radians, follow the same conversion formula:

• Use \(-225^{\circ}\) like any other degree measure.
• Apply the formula: \(-225^{\circ} \times \frac{\pi}{180} = -\frac{5\pi}{4}\).

Negative angles appear often in trigonometry and geometry, representing rotations. Once you understand the conversion, dealing with negative angles becomes straightforward. You can interpret them in the context of counterclockwise (positive) and clockwise (negative) directions.