18.5 Applications of Geometric Sums and Series 591
22. Supposeyouborrow$18,000ataninterestrateof8%compoundedannually.Youbegin paying back money four years from today and make xed payments annually. You pay back the entire debt after six payments. What are your annual payments?
Begin by guring out the ballpark gures. Will you pay more than $3000 each year? What is an upper bound for the amount of money you will pay each year?
23. A prince takes out a loan of $200,000 in order to nance his castle. The interest rate is 12% per year compounded monthly and he has a 15-year mortgage. He will pay back the loan by paying a xed amount, dollars, every month beginning one month from today and continuing for the next 15 years. What is ?
Note that the sum of the present values of his payments (pulled back to the present using an interest rate of 12%) should equal his loan.
24. Lithium,adrugthatisusedtotreatmanicdepression,orbipolardisorder,hasahalf-life of 24 hours. Suppose a patient begins taking a pill of mg every 12 hours.
What is the level of the drug in the patients body two weeks into treatment, immediately after taking the 28th pill?
25. A physician prescribes a pill to be taken daily. Suppose that the half-life of the medica-tion in the patients bloodstream is 10 days. How many milligrams of medicine should the doctor prescribe if she wants the maximum level of the drug in the bloodstream to reach mg, but not to surpass it. Assume that the drug is to be taken indenitely. Your answer will be in terms of .
26. At the beginning of each month a medical research center buries its refuse in its refuse dump. The monthly refuse deposit contains 40 grams of radioactive material. The radioactivematerialdecaysatarateproportionaltoitself,withproportionalityconstant 0.2.
(a) How much of the radioactive material buried at the beginning of the month is radioactive months later?
(b) Immediatelyafterthe60thmonthlydump, howmuchradioactivematerialisinthe refuse site?
(c) If the situation goes on indenitely, how much radioactive material will the site contain?
27. You take out a loan of $3000 at an interest rate of 6% compounded monthly. You start paying back the loan exactly one year later. How much should each payment be if the loan is paid off after 24 equal monthly payments? Give an exact answer and an approximation correct to the nearest penny.
P A R T VII
C H A P T E R
Trigonometry: Introducing Periodic Functions
Transition to Trigonometry
Familiarity with a variety of families of functions provides us with tools necessary for modeling phenomena in the world around us. In the next few chapters we will work with functionsthatareparticularlyusefulinmodelingcyclicorrepeatingphenomenabecausethe functions themselves are cyclic. Examples of cyclic behavior abound in nature; the rhythm of a heartbeat, the length of a day, the height of the sun in the sky, the path of a sound wave, and the motion of the planets all feature repeating patterns. Wheels are spinning all around uswheels of bikes, trucks, carsgears of vehicles, watches, and other machines. Think about the motion of a spot on a steadily rotating gear, or of a seat on a steadily spinning Ferriswheel.Theheightoftheseatisacyclicfunctionoftime; itrisesandfallsinasmooth, repeating manner. In this chapter we introduce trigonometric functions,1 cyclic functions that exhibit and help us explore behaviors we observe around us.
1 The word trigonometry refers to triangles, not circles. Trigonometry can be viewed in two distinct ways. Historically it developed in the context of triangles, and hence the name of this family of functions refers to triangles. We will take a triangle perspective in Chapter 20.
594 CHAPTER 19 Trigonometry: Introducing Periodic Functions
19.1 THE SINE AND COSINE FUNCTIONS: DEFINITIONS AND BASIC PROPERTIES
D e f i n i t i o n
Below is a circle of radius 1 centered at the origin. This is referred to as the unit circle. Well dene trigonometric functions with reference to a point %, on the unit circle. We locate using a real number as follows.
Start at 1, 0.
If 0, travel along the circle in a counterclockwise direction a distance units to arrive at .
If 0, travel along the circle in a clockwise direction a distance units to arrive at .
In other words, indicates a directed distance around the unit circle. Equivalently, indicatesadirectedarclengthfrom(1,0),wherethetermarclengthmeansadistance along a circle. As varies, the point moves around the unit circle. As the position of varies, so do its %- and -coordinates.
P(x) = (cos x, sin x) sin x
u cos x 1
directed distance along the circle
u2 + v2 = 1
sin x = the -coordinate of
cos x = the %-coordinate of
(the vertical coordinate, the second coordinate, the (signed) height of )
(the horizontal coordinate, the rst coordinate)2
These functions are called the sine and cosine functions, respectively.
From these denitions all the properties of the trigonometric functions follow.
2 Heres a way to remember which is which: put cosine and sine in alphabetical order. Cosine is rst; it corresponds to the rst coordinate of . Sine is second, and corresponds to the second coordinate of . We have labeled our coordinate axes % and because we want to determine the point , making sine and cosine functions of .
19.1 The Sine and Cosine Functions: Denitions and Basic Properties 595
Our method for locating essentially involves wrapping the real number line around the unit circle, with zero glued to the point (1, 0) on the circle. Figure 19.2 shows the unit circle with the portion of the number line from 0 to 2 wrapped around it like measuring tape. Well refer to this as the calibrated unit circle.
Use the calibrated unit circle to approximate sin1.1 and cos1.1.
Locate 1.1 by moving along the unit circle a distance 1.1 counterclockwise from the point 1.0. Approximate the coordinates of 1.1. 1.1 0.45, 0.9. Therefore sin1.10.9 and cos1.10.45.
v P(1.1) (.45, .9)
.3 .2 .1
.1 .2 .3
0.3 0.2 0.1
EXERCISE 19.1 (a) Find values of between 0 and 2 such that the coordinates of are (i) 1, 0. (ii) 0, 1. (iii) 1, 0. (iv) 0, 1.
(b) Use the denitions of sine and cosine to evaluate the following.
(i) sin (ii) cos (iii) sin (iv) cos (v) sin 3 (vi) cos 3
EXERCISE 19.2 Use the calibrated unit circle shown to approximate the following. (You can check your answers using a calculator, but be sure that the calculator is in radian mode as opposed to degree mode.)3
(i) sin 0.3 (ii) sin 2 (iii) sin 3 (iv) sin 4 (v) sin 5 (vi) cos 5
EXERCISE 19.3 Use the calibrated unit circle shown to approximate all -values between 0 and 2 such that
(i) sin 0.8. (ii) sin 0.4. (iii) cos 0.8. (iv) cos 0.2.
3 Radians and degrees will be discussed in Section 19.4. To check whether your calculator is in radian mode, try to evaluate cos . You will get 1 only if your calculator is in radian mode.