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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 65. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | 19.4 Angles and Arc Lengths 621 1 0 the line drawn from the origin to 0 1 makes an angle of radians with the positive horizontal axis. v v v Figure 19.22 Radians are unitless. When we write sin 1.1 and want to think of 1.1 as an angle then it is an angle in radians. In other words sin 1.1 sin 1.1 radians . Converting Degrees to Radians and Radians to Degrees You may be accustomed to measuring angles in degrees but radians are much more convenient for calculus. We need a way to go back and forth between the two measures. The fact that one full revolution is 360 or 2n radians allows us to do this. Equivalently half a revolution is 180 or n radians. 180 n radians n radians 180 degrees 1 -----radians 180 1 radian 2 degrees 57 EXERCISE 19.12 Convert from degrees to radians or radians to degrees. a 45 b -30 c 3n radians d -2 radians Answers a n b - 6 c 270 d - -114.59 Arc Length Arc lengths are usually described in terms of the angle that subtends the arc and the radius of the circle. We know that on a unit circle an angle of x radians subtends an arc length of x units. What about on a circle of radius 3 The circumference of a circle of radius 3 is three times that of the unit circle so the arc length subtended by an angle of x radians on the circle of radius 3 should be 3x. 622 CHAPTER 19 Trigonometry Introducing Periodic Functions An arc subtended by an angle of O radians on a circle of radius r will have arc length r . Notice that the formula for arc length is simple when the subtending angle is given in radian measure. The corresponding arc length formula for an angle of degrees is r ------ 180 r On 180 Trigonometric Functions of Angles We can think of the input of any trigonometric function as either a directed distance along the unit circle or as an angle because an angle in standard position will determine a point on the unit circle. EXAMPLE 19.4 Suppose angle O is in standard position and the point lies on the terminal side of O but does not lie on the unit .
