Problem 61 - Precalculus : Building Concepts and Connections

Question

Convert each angle from degrees to radians. $$390^{\circ}$$

Step-by-Step Solution

Answer

The angle 390 degrees is equal to \( \frac{13\pi}{6} \) radians.

Step 1: Identify the given angle in degrees

The angle given in the problem is 390 degrees.

Step 2: Apply the conversion factor

Multiply the given angle by the conversion factor. The conversion factor from degrees to radians is \( \frac{\pi}{180} \). Hence, the conversion will follow the formula: Angle in radians = angle in degrees * \( \frac{\pi}{180} \).\ So, we replace the angle in degrees with 390 and write down the equation for conversion: 390 degrees = 390 * \( \frac{\pi}{180} \) radians.

Step 3: Simplify

Simplify the product to obtain the measure in radians. This results in \( \frac{390\pi}{180} \) radians. Simplifying further gives \( \frac{13\pi}{6} \) radians.

Key Concepts

Degrees to Radians Conversions

Angles are typically measured in degrees, but sometimes it's useful to express them in radians instead. Understanding this conversion is key in many fields, such as physics, engineering, and mathematics. To convert from degrees to radians, you can use the formula: \( ext{Radians} = ext{Degrees} \times \frac{\pi}{180} \).
The angle given in this exercise is 390 degrees. Applying the conversion formula, we multiply 390 degrees by the fixed fraction \( \frac{\pi}{180} \), which represents the ratio of radians to degrees in a circle. The result will give us the angle in radians.
This conversion is crucial because it helps in integrating different mathematical concepts and ensures consistency in calculations across various disciplines.

Understanding the Conversion Factor

The conversion factor \( \frac{\pi}{180} \) is derived from the relationship between the circumference of a circle and its radius. This factor is constant and originates from the definition of a full circle in both radians and degrees:

• Full circle in degrees: 360 degrees
• Full circle in radians: \(2\pi\) radians

Since \(2\pi\) radians correspond to 360 degrees, dividing \(2\pi\) by 360 simplifies to \( \frac{\pi}{180} \).
This factor serves as a bridge between two different units of measuring angles, ensuring you can accurately convert between them whenever necessary. Multiplying any angle in degrees by this factor gives you the equivalent angle in radians seamlessly. Knowing this conversion factor and understanding where it comes from is essential for anyone working with trigonometry and calculus, as it influences many trigonometric identities and circle measurements.

Simplifying Fractions in Conversions

Once you've multiplied the degrees by the conversion factor, you often end up with a fraction that can be simplified. Simplification makes the resultant fraction easier to understand and work with. In this exercise, the fraction obtained was \( \frac{390\pi}{180} \).

• First, identify the greatest common divisor (GCD) of the numerator and the denominator (here it's 30).
• Then, divide both the numerator and the denominator by this GCD: \( \frac{390 \div 30}{180 \div 30} \).

This results in \( \frac{13\pi}{6} \), which is the simplest form of the fraction. Simplifying fractions is an important step that helps in reducing complexity in equations and enhances accuracy in solving mathematical problems by minimizing rounding errors.Mastering this skill is essential for working with ratios, algebraic equations, and calculus operations efficiently.