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Chapter 8: Problem 78

Convert to degree measure. $$3 \pi$$

Short Answer

Expert verified
3蟺 radians is equivalent to 540 degrees.

Step by step solution

01

Identify the Radian Measure

Understand that the given angle is in radians. In this case, the angle is given as $$3 \text{蟺}.$$
02

Understand the Conversion Factor

Recall that to convert radians to degrees, you multiply by the conversion factor $$\frac{180掳}{蟺}.$$ This is because 蟺 \text{ radians} = 180掳.
03

Set Up the Conversion

Multiply the given angle in radians by the conversion factor: $$3 蟺 \times \frac{180掳}{蟺}.$$
04

Simplify the Expression

Cancel out 蟺 from the numerator and the denominator: $$300 \times \frac{180掳}{鈭弣 = 3 \times 180掳.$$
05

Calculate the Result

Perform the multiplication: $$3 \times 180掳 = 540掳.$$Thus, $$3蟺$$ radians is equivalent to $$540掳.$$

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Key Concepts

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

angle measurement
Angles are a fundamental concept in geometry and trigonometry. They help us describe the rotation or turn around a point.
An angle is usually measured in two units: degrees and radians.
Imagine the corner where two lines meet. The amount of turn between these lines can be called an angle.
Each unit provides a different way of interpreting this turn.
Being familiar with angle measurement is crucial for understanding geometric shapes, analyzing patterns, and even in real-world applications like navigation.
Let's dive into radians and degrees to better understand these common units.
radian measure
Radians are another way to measure angles, often used in higher-level mathematics and physics. A radian is based on the radius of a circle.
One radian is the angle made when you take the radius of a circle and 'wrap' it along the circle's edge.
In simpler terms, if you straighten out the arc created by a circle's radius, the angle formed is one radian.
The complete circle has an angle of \(2 \text{蟺}\) radians. This is because it takes \(2 \text{蟺}\) radii to wrap completely around the circle.
Remember that \(\text{蟺} \text{ radians} = 180掳\).
So halfway around the circle is 蟺 radians, and all the way around is 2蟺 radians.
degree measure
Degrees are a more common way to measure angles, especially in everyday settings.
There are 360 degrees in a full circle.
To give you a sense of scale: a right angle, like the corner of a square, is 90 degrees.
This system of measuring angles can be handy for visualizing and communicating angles easily.
Converting between radians and degrees mainly involves using the fact that \(\text{蟺} \text{ radians} = 180掳\).
For example, in our exercise, we converted 3蟺 radians to degrees by using the relationship:
  • 3蟺 radians 脳 \(\frac{180掳}{蟺}\)
  • This simplifies to 3 脳 180掳
  • leading to 540掳.

This understanding will make it easier to quickly switch between radians and degrees, depending on what the problem requires.

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