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Chapter 4: Problem 15

Convert each angle to degrees. $$ -\frac{2 \pi}{3} \text { radians } $$

Short Answer

Expert verified
The angle \(-\frac{2 \pi}{3}\) radians is equal to -120掳 in degrees.

Step by step solution

01

Write down the formula

To convert the angle from radians to degrees, we'll use the following formula: degrees = radians 脳 \(\frac{180掳}{\pi}\) Our angle in radians is \(-\frac{2 \pi}{3}\).
02

Replace radians value in the formula

Now, we'll replace the radians value with our given angle: degrees = \(-\frac{2 \pi}{3}\) 脳 \(\frac{180掳}{\pi}\)
03

Simplify the expression

Next, we'll simplify the expression. Notice that the \(\pi\) in the numerator and denominator will cancel each other out: degrees = \(-\frac{2}{3}\) 脳 180掳
04

Calculate the result

Let's now calculate the final result by multiplying: degrees = -120掳 The angle \(-\frac{2 \pi}{3}\) radians is equal to -120掳 in degrees.

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Key Concepts

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

Radians
Radians are a way to measure angles based on the radius of a circle. Imagine you have a circle, and you take a piece of string and wrap it along the edge. If the length of the string equals the radius of the circle, the angle formed at the center is one radian.

  • A full circle is 2蟺 radians because the circumference of a circle is 2蟺 times the radius.
  • Radians can be both positive and negative. Negative radians indicate a clockwise rotation, while positive radians are counterclockwise.

Understanding radians is crucial when working with circular or periodic phenomena, like waves or rotations. It's a measurement that relates directly to the properties of the circle rather than being an arbitrary degree measure.
Degrees
Degrees are a more widespread and familiar way to measure angles. A full circle is divided into 360 equal parts, making a complete revolution 360 degrees.

  • The concept of dividing a circle into 360 parts comes from ancient Babylonian astronomy.
  • Like radians, degrees can be positive or negative, representing different rotation directions.
  • Degrees are versatile because they're easier to visualize and commonly used in everyday contexts, such as navigation or carpentry.

Degrees are helpful because they're intuitive, providing a clear understanding of an angle's size at a glance.
Formula
To convert between radians and degrees, you use a simple but essential formula. The relation between radians and degrees is based on the circumference of the circle.

The formula is: \[\text{degrees} = \text{radians} \times \frac{180掳}{\pi}\]

Here's how this formula works:
  • Multiply the radian measure by \(\frac{180掳}{\pi}\) to convert to degrees.
  • The \(\pi\) terms cancel out during conversion, simplifying the process.

Using this conversion formula lets you easily switch angles between these two systems. This helps in various fields like engineering and physics, where you might need to work in a specific angle measurement.

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Most popular questions from this chapter

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\cos (-v)\)

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\tan (-u)\)

Find exact expressions for the indicated quantities. \(\cos \left(\frac{\pi}{2}-u\right)\)

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\cos (u+4 \pi)\)

Pretend that you are living in the time before calculators and computers existed, and that you have a table showing the cosines and sines of \(1^{\circ}, 2^{\circ}, 3^{\circ},\) and so on, up to the cosine and sine of \(45^{\circ}\). Explain how you would find the cosine and sine of \(71^{\circ}\), which are beyond the range of your table.
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