GRAPHS OF TRIGONOMETRIC FUNCTIONS - Plainview

1 CHAPTER11434 CHAPTERTABLE OFCONTENTS11-1 Graph of the sine Function11-2 Graph of the Cosine Function11-3 Amplitude, Period, and PhaseShift11-4 Writing the Equation of aSine or Cosine Graph11-5 Graph of the TangentFunction11-6 GRAPHS of the ReciprocalFunctions11-7 GRAPHS of InverseTrigonometric Functions11-8 Sketching TrigonometricGraphsChapter SummaryVocabularyReview ExercisesCumulative ReviewGRAPHS OFTRIGONOMETRICFUNCTIONSM usic is an integral part of the lives of most peo-ple. Although the kind of music they prefer will differ,all music is the effect of sound waves on the ear.

2 Soundwaves carry the energy of a vibrating string or columnof air to our ears. No matter what vibrating object iscausing the sound wave, the frequency of the wave(that is, the number of waves per second) creates asensation that we call the pitch of the sound. A soundwave with a high frequency produces a high pitch whilea sound wave with a lower frequency produces a lowerpitch. When the frequencies of two sounds are in theratio of 2 : 1, the sounds differ by an octave and pro-duce a pleasing combination. In general, music is theresult of the mixture of sounds that are mathematicallyrelated by whole-number ratios of their is just one of many physical entities that aretransmitted by waves.

3 Light, radio, television, X-rays,and microwaves are others. The TRIGONOMETRIC func-tions that we will study in this chapter provide themathematical basis for the study of 8/12/08 1:54 PM Page 434 The sine function is a set of ordered pairs of real numbers. Each ordered paircan be represented as a point of the coordinate plane. The domain of the sinefunction is the set of real numbers, that is, every real number is a first elementof one pair of the sketch the graph of the sine function, we will plot a portion of the graphusing the subset of the real numbers in the interval 0 x 2p.

4 We know thatsin that is the measure of the reference angle for angles with measures of ,,, .. We also know thatsin ..and that is the measure of the reference angle for angles with measures of ,,, .. We can round the rational approximation of sin to two decimalplaces, the graph, we plot the points whose coordinates are given in the these points, we draw a smooth curve. Note how xand ychange. As xincreases from 0 to ,yincreases from 0 to 1. As xincreases from to p,ydecreases from 1 to 0. As xincreases from pto ,ycontinues to decrease from 0 to 21.

5 As xincreases from to 2p,yincreases from 21 to 1p32p3xy = sin xp6p25p6p7p64p33p25p311p62pp35p34p32p3p3 !32p311p67p65p6p612p611-1 GRAPH OF THE sine FUNCTIONG raph of the sine Function435x0p2psin 8/12/08 1:54 PM Page 435 When we plot a larger subset of the domain of the sine function, this pat-tern is repeated. For example, add to the points given above the point whose x-coordinates are in the interval 22p x time we increase or decrease the value of the x-coordinates by a mul-tiple of 2p, the basic sine curve is repeated. Each portion of the graph in aninterval of 2pis onecycleof the sine graph of the functiony5sin xis its own image under the translationT2p,0.

6 The function y5sin xis called a periodic functionwith a periodof 2pbecause for every xin the domain of the sine function, sin x5sin (x12p). The period of the sine function y5sin xis cycle of the sine curve can be separated into four quarters. In the firstquarter, the sine curve increases from 0 to the maximum value of the the second quarter, it decreases from the maximum value to 0. In the thirdquarter, it decreases from 0 to the minimum value, and in the fourth quarter, itincreases from the minimum value to 1Oy = sin x22p3p222pp22p2p3p22p5p23p7p2yx22p13p222 pp2221p2p3p22py 5 sin xO436 GRAPHS of TRIGONOMETRIC Functionsx22p222222p222220sin 8/12/08 1:54 PM Page 436A graphing calculator will display the graph of the sine the calculator in radian the equation for the sine display one cycle of the curve, letthe window include values from 0 to 2pfor xand values slightly smaller than21 and larger than 1 for y.

7 Use the fol-lowing viewing window:Xmin50,Xmax52p,Xscl 5, , (Note:Xscl changes thescale of the x-axis.) ENTER:0 2 6 , graph the sin curve by pressing . To display more than one cycle of the curve, change Xminor Xmaxof the :22 4 GRAPHENTERp2ndENTERp2ndWINDOWGRAPHENTERE NTERENTER p2ndENTERp2ndENTERWINDOWp6 ENTERX,T, ,nSINY ENTER MODEG raph of the sine Function437 Sci Eng0123456789 DegreeNormalFloatRadian Plot1 Plot2 Plot3\Y1 sin(X\Y2==WINDOW Xmin=0 Xmax= Xscl=.)

8 Ymin= Ymax= Yscl=1 Xres= 8/12/08 1:54 PM Page 437 EXAMPLE 1In the interval 22p x 0, for what values of xdoes y5sin xincrease and forwhat values of xdoes y5sin x decrease?SolutionThe graph shows that y5sin xincreases in the interval 22p x and in the interval x 0 and decreases in the interval x .The Graph of the sine Function and the Unit CircleRecall from Chapter 9 that if ROP is an angle in standard position with mea-sure uand P(p,q) is a point on the unit circle, then (p,q) 5(cos u, sin u) andA(u,q) is a point on the graph of y5sin x. Note that the x-coordinate of Aonthe graph of y5sin xis u, the length of.

9 Compare the graph of the unit circle and the graph of y5sin xin the fig-ures below for different values of (p, q)A(u, q)2121112pO212p223p22p223p2438 GRAPHS of TRIGONOMETRIC 8/12/08 1:54 PM Page 438 Hands-On Activity: Unwrapping the Unit CircleWe can use the graphing calculator to explore the unit circle and its relationshipto the sine and cosine . Select RADIAN mode, PAR graphing mode, andSIMUL graphing to enter the windowscreen. Use the following viewingwindow:Tmin50,Tmax52p, ,Xmin521,Xmax52p,Xscl 5, , the and arrow keys to display Yminand from Chapter 9 that a point Pon the unit circle has coordinates (cos u, sin u) where uis the measure ofthe standard angle with terminal sidethrough P.

10 We can define a function onthe graphing calculator that consists ofthe set of ordered pairs (cos u, sin u).Its graph will be the unit :This key sequence defines the function consisting of the set of orderedpairs (cos T, sin T). The variable T represents uon the graphing , we can define a function consisting of the set of ordered pairs(u, sin u).ENTER:)X,T, ,nSIN X,T, ,n )X,T, ,nSIN )X,T, ,nCOSY WINDOW Tstep=.1 Xmin=-1 Xmax= Xscl=. Ymin= Ymax= Yscl=1<WINDOW Tmin=0 Tmax= Tstep=.1 Xmin=-1 Xmax= Xscl=. Ymin= <p6 WINDOWMODEG raph of the sine Function439 Sci Eng 0123456789 Degree NormalFloatRadianFunc Pol Seq DotSequential Real re^ui Horiz G-TParFullConnectedSimula+bi Plot1 Plot2 Plot3\X1T cos(T) Y1T sin(T)\X2T T Y2T sin(T)\X3T = Y3T =\X4T ===== 8/12/08 1:54 PM Page to watch the unit circle unwrap into the sine function.