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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 63. 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 | 19.1 The Sine and Cosine Functions Definitions and Basic Properties 601 Figure 19.9 x is minimum when sin x 0 that is when x n where n is any integer. The minimum value of is 0. x is maximum where sin x takes on the values of 1. is maximum at x . 2 32 52 . In other words x is maximum at x y n where n is any integer. The maximum value of is 2. Question Let x y 4 - 4 cos2 x - as in Example 19.2. Do you think that x 0 at the local maxima of At the local minima of PROBLEMS FOR SECTION 19.1 1. Use a straightedge and the calibrated unit circle drawn below to estimate each of the following values. You can check your answers with a calculator set in radian mode. a cos 1.1 e cos 5.9 d cos 4.2 b sin 1.1 f sin 2.2 c sin 3.5 g sin 5.7 2. Use the calibrated unit circle to estimate all t-values between 0 and 6 such that a cos t 0.3. b sin t 0.7. c sin t -0.7. 602 CHAPTER 19 Trigonometry Introducing Periodic Functions 3. P w is indicated in the figure below. Find the following. a sin w b cos w c sin -w d cos -w e sin w 6 f cos w 2 g Is cos 2w positive negative or zero Explain briefly. 4. Beginning at point 1 0 and traveling a distance t counterclockwise along the unit circle we arrive at a point with coordinates 23-2 j. Find the following. a cos t b sin t c sin t d cos t e sin t f sin t 10 g Is sin t y positive negative or zero Explain. 5. Which of the following equations hold for all x Explain your answers in terms of the unit circle. a sin x sin x b sin x sin x c cos x cos x d cos x cos x 6 6. Evaluate the following limits. Explain your reasoning. a limx OT sin x b limx OT 19.2 Modifying the Graphs of Sine and Cosine 603 7. Evaluate the following limits. a limx œ cos x x2 d limx œ sin x 1 b limx 0 sin Ç J e limx œ cos Ji 3 9 c limr_ .0 J x 0 sin j 19.2 MODIFYING THE GRAPHS OF SINE AND COSINE We began our discussion of trigonometric functions by suggesting that they would be useful for modeling periodic phenomena. In order for the sine and cosine functions to be useful to us we
