Chapter8 - Pre-Calculus Chapter 8A Chapter 8A Angles and...

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© : Pre - Calculus - Chapter 8A Chapter 8A - Angles and Circles Many applications of calculus use trigonometry, which is the study of angles and functions of angles and their application to circles, polygons, and science. We start with the definition of angles and their measures. Angles Roughly, an angle is the space between two rays or line segments with a common endpoint. The rays or line segments are called the sides and the common endpoint is called the vertex. More precisely, if a ray or line segment rotates about an endpoint from some initial position, called the initial side, to some final position, called the final side, then the angle between the sides is the space swept out. Within a plane, we say the angle is positive if the rotation is counterclockwise, and is negative if the rotation is clockwise. An angle is in standard position if the vertex is at the origin and the initial side is along the positive x -axis. θ A positive angle θ x y An angle in standard θ A negative angle Measures of Angle The size of an angle may be measured in revolutions (rev), in degrees (°) or in radians (rad). An angle is called a full rotation if the ray rotates from the initial side all the way around so that the final side coincides with the initial side. A full rotation is measured as 1rev 360° 2 rad. An angle is called a half rotation or a straight angle if the ray rotates from the initial side to a final side which is directly opposite to the initial side. A half rotation is measured as 1 2 rev 180° rad. An angle is called a quarter rotation or a right angle if the ray rotates from the initial side to a final side which is perpendicular to the initial side. A quarter rotation is measured as 1 4 rev 90° 2 rad. An angle is called a null rotation if the ray never rotates so that the final side coincides with the initial side. A null rotation is measured as 0rev 0rad. © : Pre - Calculus
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© : Pre - Calculus - Chapter 8A Several other important angles are: 3 4 rev 270° 3 2 rad 1 6 rev 60° 3 rad 1 8 rev 45° 4 rad 1 12 rev 30° 6 rad Example 1 : Find the angular measure of one time zone on the surface of the earth. Solution: The earth rotates once a day or by 1rev in 24 hours. Thus a 1 hour time zone has an angular measure of 1 24 rev 15° 12 rad. An angle will be bigger than a full rotation if the ray rotates from the initial side all the way around and past the initial side again. There is no limit to the size of an angle either positive or negative. Two angles with the same initial and final sides are called coterminal and their measures must differ by an integral multiple of 1rev 360° 2 rad. Below are some pictures of various angles. In each figure, the initial sides of the angles coincide.
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Chapter8 - Pre-Calculus Chapter 8A Chapter 8A Angles and...

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