Problem 14 - Algebra and Trigonometry

Question

Find the radian measure of the angle with the given degree measure. $$ 202.5^{\circ} $$

Step-by-Step Solution

Answer

The radian measure is \( \frac{9\pi}{8} \).

Step 1: Understanding the Conversion Factor

To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180} \). This comes from the fact that \( 180^{\circ} = \pi \, \text{radians} \).

Step 2: Setting Up the Conversion Equation

Use the conversion factor to form the equation: \[ 202.5^{\circ} \times \frac{\pi}{180} \]. This equation allows us to convert degrees to radians.

Step 3: Multiplying to Find Radians

Calculate the multiplication: \[ 202.5 \times \frac{\pi}{180} = \frac{202.5\pi}{180} \].

Step 4: Simplifying the Fraction

Simplify \( \frac{202.5\pi}{180} \) by dividing both the numerator and the denominator by the greatest common divisor. Doing so, we get: \[ \frac{202.5 \div 0.5 \pi}{180 \div 0.5} = \frac{405\pi}{360} \].

Step 5: Further Simplification

Simplify \( \frac{405\pi}{360} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 45: \[ \frac{405 \div 45 \pi}{360 \div 45} = \frac{9\pi}{8} \].

Key Concepts

Degree to Radians

When we measure angles, we can use degrees or radians. Degrees are more familiar to us in everyday life, but in mathematics, radians are used often for calculations. The task here is to convert degrees into radians. Radians give us a more natural way to describe angles, especially when dealing with circles and trigonometry.
To convert an angle from degrees to radians, you multiply the degree measure by a specific factor, known as the conversion factor. This allows for seamless conversion between these two units of measure.

Angle Measurement

Angle measurement is crucial in both geometry and trigonometry. It allows us to describe the orientation or rotation of a point relative to another point, usually based on circular motion. Angles can be measured in two main units: degrees and radians.

• Degrees: This unit divides a circle into 360 equal parts. It is the more popular unit in day-to-day life.
• Radians: This unit is derived from the radius of a circle. One complete revolution around a circle equals \(2\pi\) radians. Hence, \(180^{\circ} = \pi \) radians.

By knowing both units, you can switch easily between them, giving you flexibility in mathematical operations.

Conversion Factor

The conversion factor is the key to translating degrees into radians, which is essential for many mathematical and engineering tasks. The relationship between these two units is given by the formula: \(180^{\circ} = \pi \) radians. Thus, to change degrees into radians, we use the conversion factor \( \frac{\pi}{180} \).
With this factor:

• Multiply the degree measure by \( \frac{\pi}{180} \).
• This straightforward multiplication converts your degree measure into the equivalent in radians.

This factor is, therefore, the bridge between understanding angles in everyday terms (degrees) and using them mathematically in calculations (radians).

Simplifying Fractions

Simplifying fractions is important in achieving neat and digestible results, especially when dealing with angle measurements in radians. After using the conversion factor to change degrees into radians, it often results in a fractional form that isn't fully reduced.
Here's how you simplify:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• The result will be a simpler version of the fraction, often making the radian measure easier to understand and use.

Remember that simplifying fractions isn't just about making numbers smaller—it's about achieving the clearest and most useful form for further computation or analysis.