tailieunhanh - Ebook College algebra & trigonometry (6th edition): Part 2

(BQ) Part 2 book "College algebra & trigonometry" has contents: The circular functions and their graphs, trigonometric identities and equations, applications of trigonometry, analytic geometry, further topics in algebra. | 6 The Circular Functions and Their Graphs Phenomena that repeat in a regular pattern, such as average monthly temperature, fractional part of the moon’s illumination, and high and low tides, can be modeled by periodic functions. Radian Measure The Unit Circle and Circular Functions Graphs of the Sine and Cosine Functions Translations of the Graphs of the Sine and Cosine Functions Chapter 6 Quiz Graphs of the Tangent and Cotangent Functions Graphs of the Secant and Cosecant Functions Summary Exercises on Graphing Circular Functions Harmonic Motion 591 591 31/08/16 4:53 pm 592 CHAPTER 6  The Circular Functions and Their Graphs Radian Measure ■ Radian Measure Radian Measure   We have seen that angles can be measured in degrees. In more theoretical work in mathematics, radian measure of angles is preferred. Radian measure enables us to treat the trigonometric functions as functions with domains of real numbers, rather than angles. Figure 1 shows an angle u in standard position, along with a circle of radius r. The vertex of u is at the center of the circle. Because angle u intercepts an arc on the circle equal in length to the radius of the circle, we say that angle u has a measure of 1 radian. ■ Conversions between Degrees and Radians ■ Arc Length on a Circle ■ Area of a Sector of a Circle y Radian r U 0 r u = 1 radian Figure 1 x An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. It follows that an angle of measure 2 radians intercepts an arc equal in 1 length to twice the radius of the circle, an angle of measure 2 radian intercepts an arc equal in length to half the radius of the circle, and so on. In general, if U is a central angle of a circle of radius r, and U intercepts an arc of length s, s then the radian measure