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Chapter 13: Problem 18

Which of these is the largest measure of an angle? a. arcsecond b. arcminute c. degree d. radian

Short Answer

Expert verified
d. radian

Step by step solution

01

- Understanding the Units

First, it is important to understand the units of angular measurement given in the options: arcsecond, arcminute, degree, and radian. One degree is equivalent to 60 arcminutes, and one arcminute is equivalent to 60 arcseconds. Radians measure angles based on the radius of a circle.
02

- Conversion Factors

Know the conversion factors: 1 degree = 60 arcminutes, 1 arcminute = 60 arcseconds, and 1 radian ≈ 57.2958 degrees. Use these conversion factors to compare the sizes.
03

- Comparing Units

Start by comparing radian to degree: 1 radian ≈ 57.2958 degrees, which suggests that a radian is much larger than a degree (1 radian > 1 degree). Next, recall that 1 degree = 60 arcminutes and 1 arcminute = 60 arcseconds, which makes 1 degree = 3600 arcseconds and likewise, 1 degree > 1 arcminute > 1 arcsecond.
04

- Conclusion

Since 1 radian is approximately 57.2958 degrees, it is clear that the radian is larger than the degree, arcminute, and arcsecond.

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Key Concepts

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

radian
In mathematics, a radian is a unit of angular measurement used in many areas of science. It is defined based on the radius of a circle. Specifically, one radian is defined as the angle created when the arc length is equal to the radius of the circle. Therefore, in a full circle, which is 360°, there are approximately 6.283 radians (since the circumference of a circle is 2π times the radius). This means that:
  • 1 radian ≈ 57.2958 degrees
  • π radians = 180 degrees
  • 2π radians = 360 degrees
Because radians are directly related to the properties of circles, they are often the preferred unit in higher mathematics and physics.
degree
A degree is one of the most commonly used units for measuring angles. A circle is divided into 360 degrees, making it easy to express small fractions of a circle. Degrees are often subdivided into smaller units:
  • 1 degree = 60 arcminutes (60')
  • 1 arcminute = 60 arcseconds (60'')
Degrees are more intuitive than radians for many, especially those in fields like geography and navigation where angles are frequently measured. For example, a right angle has 90 degrees, and a straight angle has 180 degrees.
arcminute
An arcminute is a smaller unit of angular measurement that is equal to one-sixtieth of a degree. It is often denoted by the symbol ('). Arcminutes are used where a more precise measurement is needed. Since one degree is divided into 60 arcminutes, you can think of each arcminute as:
  • 1 arcminute = 1/60 degree
  • 60 arcminutes = 1 degree
Arcminutes are particularly useful in fields like astronomy, where precise measurements of angular distances between celestial objects are crucial.
arcsecond
An arcsecond is even a smaller unit of angular measurement. It is equal to one-sixtieth of an arcminute, or one-three thousand six-hundredth of a degree. The symbol for arcsecond is (''). Specifically:
  • 1 arcsecond = 1/60 arcminute
  • 1 arcsecond = 1/3600 degree
Arcseconds are extremely useful in applications requiring fine angular resolution, such as detailed astronomical observations and precise engineering measurements.

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Most popular questions from this chapter

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Star A and star B are two stars nearly the same distance from Earth. Star A is blue, and star \(\mathrm{B}\) is red, but they have equal brightness. Which of the following statements is true? a. Star A is more luminous than star B. b. Star A is larger than star B. c. Star A is smaller than star B. d. Star A is less luminous than star B.
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