I. TRIGONOMETRIC FUNCTIONS AND IDENTITIES
In this section we shall review the definitions and basic properties of trigonometric functions.
We shall be concerned with angles in the plane that are generated by rotating a ray (or half-line) about its endpoint. The starting position of the ray is called the initial sideof the angle and the final position of the ray is called the terminal side. The initial and terminal sides meet at a point called the vertex of the angle (Figure B.1).
An angle is considered positive if it is generated by a counterclockwise rotation and negative if it is generated by a clockwise rotation. In a rectangular coordinate system, an angle is said to be in standard position if itsvertex is at the origin and its initial side is along the positive x-axis (Figure B.2). As shown
in Figure B.3, an angle may be generated by making more than one complete
The size of an angle is commonly measured in degrees. One degree (written 1[pic]) is the measure of an anglegenerated by a ray that rotates [pic] of one revolution. Thus, there are 360[pic] in an angle of one complete revolution, 180[pic] in an angle one-half revolution, 90[pic] in an angle of one-quarter revolution, and so forth (Figure B.4).
Degrees are divided into sixty equal parts, called minutes, and minutes are divided into sixty equal parts, called seconds. Thus, one minute (written 1’) is [pic]of a degree, and one second (written 1’’) is [pic] of a minute. Smaller subdivisions of a degree are expressed as fractions of a second.
In calculus, angles are measured in radians rather than in degrees because it simplifies many important formulas. To define the radian measure of an angle, suppose that a circle of radius 1 is constructed so that its center is at the vertex of theangle. As the angle is generated by a ray rotating from the initial side to the terminal side, the intersection of this ray with the unit circle travels some distance d along the unit circle (Figure B.5). To distinguish between clockwise and counterclockwise rotations,
we introduce the signed distance or signed arc length s traveled by the point;
it is defined bys = d if the rotation is counterclockwise
s = -d if the rotation is clockwise
s = 0 if there is no rotation
The signed arc length s is called the radian measure of the angle generated by the rotating ray. Since a circle of radius 1 has a circumference of [pic], there are [pic] radians in an angle of one completerevolution, [pic] radians in an angle of one-half revolution, [pic] radians in an angle of one-quarter revolution, and so forth (Figure B.6).
The following table shows the relationship between the degree measure and radian measure of some of the more important positive angles.
|DEGREES |30[pic] 45[pic] 60[pic] 90[pic] 120[pic] 135[pic] 150[pic] 180[pic] |
||270[pic] 360[pic] |
| |[pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] |
|RADIANS |2[pic] |
Observe that in the foregoing table the degree symbol was used for angles measured in degrees, but that theangles measured in radians were given as numbers with no units specified. This is standard practice in mathematics- when no units are specified for an angle, it is understood that the units are radians. Engineers are usually more specific; they use the symbol rad to denote radians. Thus, an angle of 2[pic] radians can be denoted by 2[pic]rad or by 2[pic].
From the fact that an angle of [pic]...