Problem 35 - Precalculus

Question

Convert from radians to degrees. $$\frac{5 \pi}{12}$$

Step-by-Step Solution

Answer

\(\frac{5\pi}{12}\) radians is equal to 75 degrees.

Step 1: Understand the Conversion Factor

To convert from radians to degrees, we use the conversion factor \(180^\circ/\pi\). This factor is crucial as it allows the transition from one measurement unit to another.

Step 2: Set Up the Conversion Equation

Multiply the given radian value \(\frac{5\pi}{12}\) by the conversion factor \(\frac{180^\circ}{\pi}\). This will transform the radian measure to degrees:\[\frac{5\pi}{12} \times \frac{180^\circ}{\pi}\]

Step 3: Simplify the Expression

Cancel out \(\pi\) from the numerator and the denominator:\[\frac{5}{12} \times 180^\circ\]Next, multiply \(\frac{5}{12}\) by 180:

Step 4: Perform the Multiplication

Carry out the multiplication:\[\frac{5 \times 180}{12} = \frac{900}{12}\]

Step 5: Simplify the Fraction

Divide 900 by 12 to simplify:\[\frac{900}{12} = 75\]

Step 6: Conclusion

The result of the conversion from \(\frac{5\pi}{12}\) radians to degrees is 75 degrees.

Key Concepts

Angle Measurement

In the world of geometry, angle measurement is a fundamental concept that helps us describe how much one line rotates or twists around a fixed point relative to another line. There are two main units of measuring angles: degrees and radians.

• Degrees are often used in everyday scenarios, such as when we talk about the angles in a triangle or on a protractor.
• Radians, on the other hand, are a more mathematical way to measure angles, especially handy in calculus and trigonometry. A radian measures the angle created when the radius of a circle is wrapped along its circumference. In other words, it's the ratio of the length of an arc to its radius.

Understanding how to convert between these two units is essential. This is where the conversion factor plays its role.

Conversion Factor

A conversion factor is a mathematical tool that allows us to change a measurement from one unit to another. When switching from radians to degrees, we rely on the specific relationship between these two units. The essential conversion factor to remember is that \[ 1 ext{ radian} = \left( \frac{180^\circ}{\pi} \right) \] This means every radian can be transformed into degrees by multiplying by \(\frac{180^\circ}{\pi}\).Using this conversion factor:

• We can take any angle measured in radians and turn it into an equivalent degree measurement.
• It ensures that the angle retains its true size, just in a different measurement system.

This understanding helps us set up the conversion, ensuring each step reflects real-world logic.

Simplifying Fractions

Simplification is a process that makes fractions easier to understand and work with by reducing them to their simplest form. When converting radians to degrees, simplifying fractions is a crucial final step. Let's apply this to our ongoing example:Once you multiply \[ \frac{5}{12} \times 180 \] You get \( \frac{900}{12} \). Simplifying this fraction means finding the greatest common divisor of the numerator and denominator and dividing both by it.

• In our case, 12 is the common divisor.
• So, divide 900 by 12, resulting in 75.

The result, 75, is the angle in degrees. Simplifying the expression ensures the final answer is in the most understandable and practical form.