Circles in the Plane Help (page 2)

By — McGraw-Hill Professional
Updated on Aug 30, 2011

Degrees, Minutes, Seconds

The angular degree (°), also called the degree of arc, is the unit of angular measure most familiar to lay people. One degree (1°) is 1/360 of a full circle. An angle of 90° represents a quarter circle, 180° represents a half circle, 270° represents a three-quarter circle, and 360° represents a full circle. A right angle has a measure of 90°, an acute angle has a measure of more than 0° but less than 90°, and an obtuse angle has a measure of more than 90° but less than 180°.

To denote the measures of tiny angles, or to precisely denote the measures of angles in general, smaller units are used. One minute of arc or arc minute, symbolized by an apostrophe or accent (′) or abbreviated as m or min, is 1/60 of a degree. One second of arc or arc second, symbolized by a closing quotation mark (″) or abbreviated as s or sec, is 1/60 of an arc minute or 1/3600 of a degree. An example of an angle in this notation is 30° 15′ 0″, which denotes 30 degrees, 15 minutes, 0 seconds.

Alternatively, fractions of a degree can be denoted in decimal form. You might see, for example, 30.25°. This is the same as 30° 15′ 0″. Decimal fractions of degrees are easier to work with than the minute/second scheme when angles must be added and subtracted, or when using a conventional calculator to work out trigonometry problems. Nevertheless, the minute/second system, like the English system of measurements, remains in widespread use.

Circles in the Plane Practice Problems

Practice 1

A text discussion tells you that θ 1 = π/4. What is the measure of θ 1 in degrees?

Solution 1

There are 2π rad in a full circle of 360°. The value π/4 is equal to 1/8 of 2π. Therefore, the angle θ 1 is 1/8 of a full circle, or 45°.

Practice 2

Suppose your town is listed in an almanac as being at 40° 20′ north latitude and 93° 48′ west longitude. What are these values in decimal form? Express your answers to two decimal places.

Solution 2

There are 60 minutes of arc in one degree. To calculate the latitude, note that 20′ = (20/60)° = 0.33°; that means the latitude is 40.33° north. To calculate the longitude, note that 48′ = (48/60)° = 0.80°; that means the longitude is 93.80° west.

Practice Problems for these concepts can be found at:  The Circle Model Practice Test

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