Problem 7 - Precalculus

Question

Convert to radian measure. Express your answers both in terms of \(\pi\)and as decimal approximations rounded to two decimal places. (a) \(0^{\circ}\) (b) \(360^{\circ}\) (c) \(450^{\circ}\)

Step-by-Step Solution

Answer

0° = 0 radians; 360° = 2π ≈ 6.28 radians; 450° = 5π/2 ≈ 7.85 radians.

Step 1: Understand the Conversion Formula

To convert degrees to radians, we use the formula \( \text{radians} = \frac{\pi}{180} \times \text{degrees} \). We will apply this formula to each angle measurement given in degrees.

Step 2: Convert 0° to Radians

Apply the conversion formula to 0°: \[ \text{radians} = \frac{\pi}{180} \times 0 = 0 \pi = 0 \]. Therefore, 0° in radians is 0, which is the same for both forms.

Step 3: Convert 360° to Radians

Apply the conversion formula to 360°: \[ \text{radians} = \frac{\pi}{180} \times 360 = 2\pi \]. As a decimal approximation, this is approximately 6.28 radians.

Step 4: Convert 450° to Radians

Apply the conversion formula to 450°: \[ \text{radians} = \frac{\pi}{180} \times 450 = \frac{5\pi}{2} \]. As a decimal approximation, this is approximately 7.85 radians.

Key Concepts

Degree to Radian Formula

When converting angle measurements from degrees to radians, the essential formula to use is \( \text{radians} = \frac{\pi}{180} \times \text{degrees} \). This formula takes the degree measurement and scales it by a factor of \( \frac{\pi}{180} \), which is derived from the fact that a full circle is 360 degrees or \( 2\pi \) radians. Hence, \( \pi \) radians correspond to 180 degrees, which is the core reason behind the formula.

• Degrees: Measure based on dividing a circle into 360 equal parts.
• Radians: Based on the arc length of a circle, where one radian is the angle with an arc length equal to the radius.

To easily remember: Multiply the degree by \( \frac{\pi}{180} \) to find the equivalent measure in radians. Don't forget this conversion factor, as it anchors degree values into the radian scheme.

Radian Approximation

When you convert degrees to radians, the results are often expressed in two forms: exact and approximate. The exact form incorporates \( \pi \), while the approximate form gives a more numerically friendly version with decimal places.To understand radian approximation:

• Exact Form: Results are expressed in terms of \( \pi \). For instance, 360° converts to \( 2\pi \) radians.
• Approximate Form: Use the value of \( \pi \approx 3.14159 \) to find a decimal equivalent. For example, \( 2\pi \approx 6.28 \).

Approximating radians helps in practical applications where numeric calculations are required. Thus, knowing both forms is beneficial for different mathematical tasks.

Angle Conversion Process

The process of converting an angle from degrees to radians is systematic and involves a few straightforward steps. Let's take a closer look at this essential conversion process:

• Identify the Degree Measurement: Start by knowing the degrees you want to convert, like 0°, 360°, or 450°.
• Apply the Degree to Radian Formula: Use \( \text{radians} = \frac{\pi}{180} \times \text{degrees} \). For example, converting 450° involves calculating \( \frac{\pi}{180} \times 450 = \frac{5\pi}{2} \).
• Express Answer in Exact Form: Keep \( \pi \) in the result for precision, e.g., \( \frac{5\pi}{2} \).
• Find Decimal Approximation: Convert through approximating \( \pi \) to decimals, like \( \frac{5\pi}{2} \approx 7.85 \).

This dual-step approach ensures comprehensive understanding and application, letting students tackle both theoretical and applied mathematics with confidence.