math 1330 4.2.pdf

# math 1330 4.2.pdf - Math 1330 Section 4.2 Radians Arc...

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1 Math 1330 - Section 4.2 Radians, Arc Length, and Area of a Sector The word trigonometry comes from two Greek roots, trigonon, meaning “having three sides,” and meter, meaning “measure.” We have already defined the six basic trigonometric functions in terms of a right triangle and the measures of its three sides. Before beginning our study of trigonometry, we need to take a look at some basic concepts having to do with angles. An angle is formed by two rays that share a common endpoint, called the vertex of the angle. One ray is called initial side of the angle, and the other side is called the terminal side. For ease, we typically will draw angles in the coordinate plane with the initial side along the positive x axis. We measure angles in two different ways, both of which rely on the idea of a complete revolution in a circle. You are probably familiar with degree measure. In this system of angle measure, an angle which is one complete revolution is 360°. So one degree is 1 360 th of a circle.

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2 The second method is called radian measure . One complete revolution is 2 . Suppose we draw a circle and construct an angle by drawing rays from the center of the circle to two different points on the circle in such a way that the length of the arc intercepted by the two rays is the same as the radius of the circle. The measure of the central angle thus formed is one radian. = 1 arc length = r r r
3 Radian measure of an angle: Place the vertex of the angle at the center of a circle of radius r . Let s denote the length of the arc intercepted by the angle. The radian measure of the angle is the ratio of the arc length s to the radius r . That is, s r . In general, the radian measure of a central angle can be determined by the formula s r , where s is the length of the intercepted arc and r is the radius of the circle and r and s are measured in the same units.

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