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Chapter 4: Problem 21

Convert each angle in radians to degrees. $$\frac{\pi}{2}$$

Short Answer

Expert verified
The degree measure equivalent of \( \frac{\pi}{2} \) radian is 90 degrees.

Step by step solution

01

Identification

Identify the given radian measure, which is \( \frac{\pi}{2} \) radian.
02

Conversion to Degrees

To convert radian measure to degree measure, multiply the given radian measure by \( \frac{180}{\pi} \). This means \( \frac{\pi}{2} \times \frac{180}{\pi} \).
03

Simplify

Simplify the expression. Here, \( \pi \) from the numerator and \( \pi \) from the denominator cancel out leaving \( \frac{1}{2} \times 180 \) = 90.

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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 of measuring angles based on the circumference of a circle. Think of a circle: its total circumference is given by the formula \( 2\pi \) times the radius. When you say an angle is in radians, you're essentially describing how much of the circle's circumference the angle's arc covers.

A full circle, or 360 degrees, is equal to \( 2\pi \) radians. This means that if you divide a circle into two equal parts, or a semicircle, you would have \( \pi \) radians. One important aspect to grasp here is that radians provide a natural connection to the radius of the circle because their definition is based on the circle's geometry.

To summarize:
  • 1 full rotation = \( 2\pi \) radians
  • \( \pi \) radians = 180 degrees
  • Radians relate directly to the circle's radius
Degrees
Degrees are a unit of measurement for angles that most people are familiar with. Imagine a circle divided into 360 equal parts; each part is known as a degree. This system is believed to have originated because 360 is divisible by several numbers, making calculations more manageable in ancient times.

In the degree system, angles are measured as fractions of a full circle. For instance, a right angle, which is a quarter of a circle, measures 90 degrees. When working with problems involving angles, converting back and forth between degrees and radians might be necessary, especially in mathematical calculations involving trigonometric functions.

Key points to remember:
  • 360 degrees make a full circle
  • 90 degrees make a quarter turn (right angle)
  • Common in daily life and practical applications
Conversion Formula
To convert between radians and degrees, a conversion formula is crucial. This formula stems from the relationship between radians and degrees. As mentioned, \( \pi \) radians equal 180 degrees. Using this relationship, we derive the conversion factor.

Here's how it works: when converting radians to degrees, you multiply the radian measure by \( \frac{180}{\pi} \). Conversely, when you convert degrees to radians, you multiply the degree measure by \( \frac{\pi}{180} \).

For example, to convert \( \frac{\pi}{2} \) radians to degrees, you'd calculate:
  • \( \frac{\pi}{2} \times \frac{180}{\pi} = 90 \text{ degrees} \)
Remember, the choice of using radians or degrees largely depends on the context of the problem or application at hand.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I convert degrees to radians, I multiply by \(1,\) choosing \(\frac{\pi}{180^{\circ}}\) for 1

Have you ever noticed that we use the vocabulary of angles in everyday speech? Here is an example: My opinion about art museums took a \(180^{\circ}\) turn after visiting the San Francisco Museum of Modern Art. Explain what this means. Then give another example of the vocabulary of angles in everyday use.

What is the range of the sine function? Use the unit circle to explain where this range comes from.

Determine the amplitude and period of \(y=10 \cos \frac{\pi}{6} x\) (GRAPH CANT COPY)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using radian measure, I can always find a positive angle less than \(2 \pi\) coterminal with a given angle by adding or subtracting \(2 \pi\)
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