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Chapter 5: Problem 112

Convert the radian measure \(7 \pi / 4\) to degrees.

Short Answer

Expert verified
315 degrees.

Step by step solution

01

Understand the conversion factor

To convert radians to degrees, use the conversion factor where 1 radian equals \( \dfrac{180}{\text{蟺}} \) degrees.
02

Set up the conversion calculation

Multiply the given radian measure by the conversion factor: \( \dfrac{7 \pi}{4} \times \dfrac{180}{\text{蟺}} \).
03

Simplify the expression

Cancel out the \( \text{蟺} \) terms: \( \dfrac{7}{4} \times 180 \).
04

Perform the multiplication

Multiply 7 by 180 and then divide by 4: \( \dfrac{1260}{4} = 315 \).
05

State the result

The radian measure \( \dfrac{7 \pi}{4} \) is equivalent to 315 degrees.

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Key Concepts

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

radian measure
Radians are a way to measure angles. They are based on the arc length of a circle. Think of a circle with a radius of 1 unit. One radian is the angle where the arc length equals the radius. Since the circumference of a whole circle is \(2\text{蟺}\), that means \(2\text{蟺}\) radians is 360 degrees. This is where the radian measurement connects to degree measurement.
degree measure
Degrees are a more common way to measure angles that most people are familiar with. A full circle is divided into 360 degrees. This system is easier to use for everyday situations, like telling time on a clock face or measuring angles in geometry. Degrees are a conventional unit of angular measure that dates back thousands of years. One degree is 1/360th of a full circle, making it simpler for regular use.
conversion factor
To switch between radians and degrees, you need a conversion factor. The important relationship is: 1 radian = \(\dfrac{180}{\text{蟺}}\) degrees. This comes from the fact that \(2\text{蟺}\) radians equal 360 degrees, so simplifying this gives us the conversion factor. When you multiply a radian measure by \(\dfrac{180}{\text{蟺}}\), you convert it to degrees. This factor helps us easily swap between these two ways of measuring angles.
angle conversion
Let's practice converting an angle. Take the angle \(\dfrac{7\pi}{4}\) radians and convert it to degrees. First, multiply by the conversion factor: \(\dfrac{7\pi}{4} \times \dfrac{180}{\text{蟺}}\).
Next, simplify: cancel out \(\text{蟺}\) to get \(\dfrac{7}{4} \times 180\).
Then, multiply 7 by 180 to get 1260.
Finally, divide by 4 to get \(\dfrac{1260}{4} = 315\) degrees.
So, \(\dfrac{7\pi}{4}\) radians equals 315 degrees. This process shows how using a conversion factor can help switch between radians and degrees easily.

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Most popular questions from this chapter

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