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Precalculus: Trigonometric Functions

Angles

Terms and Formulae

Problems

An angle can be defined in terms of the rotation of a single ray. If a ray rotates about its endpoint from its initial position to a second fixed position, its terminal position, these two positions are the initial and terminal sides of the angle, respectively. The endpoint of the ray is the vertex of the angle. If the rotation is counterclockwise, the angle is positive, and if the rotation is clockwise, the angle is negative.

Figure %: An angle can be generated by the rotation of a ray about its endpoint.

An angle in standard position is an angle whose vertex is at the origin (of the coordinate plane) and whose initial side lies along the positive x-axis. Angles in standard position whose terminal sides coincide are called coterminal angles (they are not necessarily equal because one could be negative and the other positive).

Figure %: Angles in standard position

There are three units with which to measure angles: revolutions, radians, and degrees. One revolution is the measure of an angle that rotates until its terminal side meets its initial side. 1 revolution = 2Π radians = 360 degrees. 360 degrees = 360 o . A degree is an arbitrary unit of measurement. A radian is not -- it occurs naturally. A radian is the measure of the central angle (of a circle) that intercepts an arc of the same length as the radius (of the circle).

The coordinate plane is divided into four regions, or quadrants. The region in which x > 0 , y > 0 is Quadrant I. The region in which x < 0 , y > 0 is Quadrant II. The region in which x < 0 , y < 0 is Quadrant III. The region in which x > 0 , y < 0 is Quadrant IV. An angle can be described as a first/second/etc. quadrant angle depending on which quadrant contains its terminal side. If 0 o < θ < 90 o , θ is a first quadrant angle. If 90 o < θ < 180 o , θ is a second quadrant angle. If 180 o < θ < 270 o , θ is a third quadrant angle. If 270 o < θ < 360 o , θ is a fourth quadrant angle. An angle whose terminal side does not lie in a specific quadrant, but instead lies along one of the axes, is called a quadrantal angle.

Figure %: The four quadrants

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