Measure of Angles

    Angles can be measured in revolutions, degrees, and radians. This topic defines degrees and radians and shows how to convert between them. Angles measured in degrees can also be repesented in decimal notaton and degree-minute-second notation. Examples are given for each representation.

Definition (Measure of Angles) Degrees. One complete revolution generates an angle with measure 360 degrees and is denoted by Thus, revolution.

Example (Measure of Angles) Quadrantal Angles in Degrees. Some quadrantal angles are: and   

Definition (Measure of Angles) Acute and Obtuse Angles. An acute angle is an angle measuring less than but more than An obtuse angle is an angle measuring more than but less than

Definition (Measure of Angles) Complementary and Supplementary Angles. Complementary angles are two positive angles whose sum is Supplementary angles are two positive angles whose sum is

Example (Measure of Angles) Special Angles) The angles and are acute and obtuse, respectively. The angles and are complementary because The angles and are supplementary because

Definition (Measure of Angles) Minutes and Seconds. One degree is sixty minutes, and is denoted by Also, one minute is sixty seconds and is denoted by

Example (Measure of Angles) Minutes and Seconds. Convert to decimal degrees.
    Solution. Since and we have

Example (Measure of Angles) Minutes and Seconds.  Find the angle that is complementary to  

     Solution. We do this by subtracting, However to do this we need to borrow, so we write Thus, we find

  

So and are complementary angles.

Definition (Measure of Angles) Radian. If an angle has its vertex at the center of a circle, it will intercept an arc length on that circle. If an angle intercepts an arc length just equal to the radius of the circle, then we have an angle subtended which measures one radian.

Proposition (Measure of Angles) Arc Length. If an angle has its vertex  at the center of a circle (central angle), then the arc length divided by the radius is equal to the number of radians in the angle; that is,

    It is important to remember that can only be used when is in radians.

Example (Measure of Angles) Arc Length. A circular spool 8.0 in. in diameter has 650 ft of thin wire around its outer rim. Through what angle will this spool turn as all the wire is pulled off?

    Solution. We will use the formula after we solve for Also we first need to have and in the same units so Thus,

    To convert from degrees to radians multiply by and to convert from radians to degrees multiply by since is 180° which is radians.

Example (Measure of Angles) Converting Between Degrees and Radians. Convert to degrees by multiplying by and conversely, to convert say 165° to radians multiplying by π/180° to get   Try the following:

Example (Measure of Angles) Unit Circle. The unit circle is the circle with radius 1 and has equation The special angles that are used quite often in trigonometry both in degrees and radians are displayed on the unit circle centered at the origin:

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Cite this as:
Measure Of Angles
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/measure-of-angles.html