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Chapter 6: Problem 25

Convert to radian measure. Round the answer to two decimal places. $$117.8^{\circ}$$

Short Answer

Expert verified
117.8 degrees is approximately 6.45 radians.

Step by step solution

01

Understand the Conversion Factor

To convert degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This means that 180 degrees is equivalent to \( \pi \) radians.
02

Set Up the Conversion

Multiply the given degree measure by \( \frac{\pi}{180} \) to convert it to radians: \[ 117.8^{\circ} \times \frac{\pi}{180} \]
03

Perform the Multiplication

Calculate the multiplication: \[ 117.8 \times \frac{\pi}{180} = 2.055 \pi \]
04

Round the Answer

Since \( \pi \) is approximately 3.14, multiply the result by \( 3.14 \).Finally, round to two decimal places: \[ 2.055 \times 3.14 \approx 6.45 \] So, \[ 117.8^{\circ} \approx 6.45 \text{ radians} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

angle measurement
Angle measurement is pivotal in trigonometry and geometry. The most common units are degrees and radians. Degrees are small sections of a circle, whose total adds up to 360 degrees in a full circle. Radians, on the other hand, are based on the radius of the circle. One complete revolution around a circle is equal to \( 2\pi \) radians. This means \( \pi \) radians is equivalent to 180 degrees. Understanding how to convert between these units is essential for solving various mathematical problems involving angles.
radian measure
Radians are another way to measure angles. One radian is the angle created when the radius wraps around the circle's edge exactly once. The circumference of a circle can be expressed using the formula \( 2\pi r \). Since the radian is defined with respect to the radius, it provides a natural and direct way of describing angles, especially in higher mathematics and physics. Remember, \( \pi \) radians equal 180 degrees, facilitating easy conversion between radians and degrees.
conversion factor
To convert degrees to radians, we use a simple conversion factor: \( \frac{\pi}{180} \). This factor comes from the relationship that 180 degrees is equal to \( \pi \) radians. Here is a step-by-step method:
  • Multiply the degree measure by \( \frac{\pi}{180} \).
  • Perform the multiplication to get the radian measure.
  • Sometimes, for practical purposes, you may need to multiply the result by the approximate value of \( \pi \) (3.14) and round to a specified decimal place.
Using this method will ensure accurate conversion from degrees to radians, as illustrated in the conversion of 117.8 degrees to approximately 6.45 radians.

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