Problem 39 - Precalculus

Question

Convert from radians to degrees. $$\frac{19 \pi}{20}$$

Step-by-Step Solution

Answer

The angle \( \frac{19\pi}{20} \) radians converts to 171 degrees.

Step 1: Identify the Conversion Factor

We know that 180 degrees is equal to \( \pi \) radians. Therefore, the conversion factor to go from radians to degrees is \( \frac{180}{\pi} \).

Step 2: Set Up the Conversion Equation

Use the conversion factor in an equation where the angle in radians, \( \theta = \frac{19\pi}{20} \), will be multiplied by \( \frac{180}{\pi} \). This gives: \[ \theta (\text{in degrees}) = \frac{19\pi}{20} \times \frac{180}{\pi} \]

Step 3: Simplify the Equation

Cancel out \( \pi \) in the numerator and the denominator: \[ \theta (\text{in degrees}) = \frac{19 \times 180}{20} \]

Step 4: Perform the Multiplication and Division

Calculate \( 19 \times 180 = 3420 \), then divide by 20: \[ \theta (\text{in degrees}) = \frac{3420}{20} = 171 \]

Step 5: Interpret the Result

The conversion of \( \frac{19\pi}{20} \) from radians to degrees gives you 171 degrees.

Key Concepts

Angle Measurement

Angles can be measured in different units, primarily degrees and radians. Degrees are in common use in daily life, often showing up in tasks like telling time or setting angles with protractors. A full circle is 360 degrees, which can be a familiar benchmark for many. On the other hand, radians are essential in mathematics and engineering.
A radian measures angles differently; it is based on the radius of a circle. In simple terms, one radian is the angle at the center of a circle when the arc that the angle subtends is equal in length to the radius of the circle.

• 360 degrees is equivalent to 2π radians.
• 180 degrees equals π radians.

Understanding how these units relate is fundamental when converting between degrees and radians.

Conversion Factor

The conversion between radians and degrees utilizes a straightforward conversion factor. Since we know that 180 degrees equals π radians, we can derive two primary conversion factors depending on the direction of the conversion:

• From radians to degrees: Multiply by \( \frac{180}{\pi} \)
• From degrees to radians: Multiply by \( \frac{\pi}{180} \)

In our original exercise, the conversion factor \( \frac{180}{\pi} \) is used because we are converting from radians to degrees. Employing this conversion factor carefully ensures that computations are accurate and straightforward to perform. Remember, the key here is to multiply the radian value by this factor to achieve the equivalent in degrees.

Radian and Degree Relationship

The relationship between radians and degrees is based on the circumference of a circle. Consider a circle with a radius of one unit:

• The full circle's circumference is \( 2\pi \) units.
• A half circle, therefore, is \( \pi \) units in circumference.

Thus, the radian measure relates directly to the length of this arc, fundamental in understanding this unit. Each degree is a small fraction of the full rotation (360), while each radian is a segment of the circle’s radius fitting into the circle’s outline. This intrinsic link showcases that radians are particularly useful when dealing with circles or periodic functions in mathematics.
The conversion process respects this relationship, bridging everyday use with sophisticated mathematical applications. By understanding this, you become adept at switching between these units seamlessly.