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1、Unit 4:Trigonometric Functions,Lesson 1:The Graphs of sin,cos and tan,Trigonometry,In grade 10 you were introduced to trigonometry by applying it to right trianglesIn grade 11 you used trigonometry to solve oblique triangles(triangles without a right angle)This required you to use sin,cos and tan fo
2、r angles greater than 90Next,you created graphs of sin and cosKnown as trigonometric functionsIn grade 12,we will create graphs of sin,cos and tan for angles between 0 and 2We now look at the trigonometric functions in radians,Graphs of sin and cos,The graphs of f(x)=sin x and f(x)=cos x when x is i
3、n degrees are:,Or if we extend them beyond 0 and 360:,Terminology,The functions f(x)=sin x and f(x)=cos x are periodicThey have a repeating patternThe period is the horizontal length of the repeating patternThe axis of curve is equation of the horizontal line that cuts the graph in halfThe amplitude
4、 is vertical distance from the axis of curve to the maximum(or minimum)pointBecause it is a distance,the amplitude is always positive,Properties of the Graph of sin,One period,Axis of curve,Amplitude,Maximum,Minimum,Period:,Axis of curve:,Maximum:,Minimum:,Amplitude:,360,The line y=0,1,-1,1,Properti
5、es of the Graph of cos,One period,Axis of curve,Amplitude,Maximum,Minimum,Period:,Axis of curve:,Maximum:,Minimum:,Amplitude:,360,The line y=0,1,-1,1,Example 1,Use your TI-83 or“Graph”to create a graph of f(x)=sin x where x is in radiansFrom the graph,determineThe periodThe axis of curveThe maximum
6、and minimum valuesThe amplitude,Example 1:Solution,One period,Axis of curve,Amplitude,Maximum,Minimum,Period:,Axis of curve:,Maximum:,Minimum:,Amplitude:,2,The line y=0,1,-1,1,Example 1:Notes,The graph of f(x)=sin x has the same shape and properties when x is in radians as it does when x is in degre
7、es:The only difference is the period360 for x in degrees2 for x in radiansThis makes sense because the only thing that changed was the units for x and the period depends on x,360,Example 2,Use your TI-83 or“Graph”to create a graph of f(x)=cos x where x is in radiansFrom the graph,determineThe period
8、The axis of curveThe maximum and minimum valuesThe amplitude,Example 2:Solution,One period,Axis of curve,Amplitude,Maximum,Minimum,Period:,Axis of curve:,Maximum:,Minimum:,Amplitude:,2,The line y=0,1,-1,1,Example 2:Notes,The graph of f(x)=cos x has the same shape and properties when x is in radians
9、as it does when x is in degrees:The only difference is the period360 for x in degrees2 for x in radiansThis is exactly what we saw for f(x)=sin x,360,Example 3,Use your TI-83 or“Graph”to create a graph of f(x)=tan x where x is in radiansFrom the graph,determineThe periodThe axis of curveThe maximum
10、and minimum valuesThe amplitude,Example 3:Solution,One period,Axis of curve,Period:,Axis of curve:,Maximum:,Minimum:,Amplitude:,The line y=0,none,none,none,The graph of tan has vertical asymptotes!,Example 3:Notes,Although it is periodic(period=),the graph of f(x)=tan x looks nothing like f(x)=sin x
11、 or f(x)=cos xf(x)=tan x has no amplitude because it has no maximum or minimum valuesf(x)=tan x has asymptotes at odd multiples ofi.e.,Why f(x)=tan x Has Asymptotes,Using the quotient identity,we can see that f(x)=tan x is a rational function:So,f(x)=tan x will have asymptotes wherever cos x=0Becaus
12、e cos x=0 when f(x)=tan x has asymptotes when,Example 4,(a)On the same axis,graph f(x)=sin(x)f(x)=sin(x)+2f(x)=sin(x)3 Make sure x is in radians(b)Describe what is happening,Example 4:Solution,The graph of f(x)=sin x is moving up and down,Example 4:Notes,By adding a value c,to f(x)=sin x,we move the
13、 functionUp when c 0 Down when c 0This is known as a vertical translationThe value of c is added to the y-coordinate of every point on the graph of f(x)=sin x Changes the axis of curve to the line y=c,Example 5,(a)On the same axis,graph f(x)=sin(x)f(x)=2sin(x)f(x)=0.5 sin(x)f(x)=-3sin(x)Make sure x
14、is in radians(b)Describe what is happening,Example 5:Solution,The graph of f(x)=sin x is being stretched and compressed.,The graph of f(x)=sin x is“flipped”over the x-axis and stretched,Example 5:Notes,By multiplying f(x)=sin x by a value a weStretch the function when a 1Compress the function when 0
15、 a 1This is known as a vertical dilationWe also reflect the function over the x-axis when a 0 This is known as a vertical reflection In both cases,The y-coordinate of every point on the graph of f(x)=sin x is multiplied by aThe amplitude is changed to|a|,Means“absolute value”and you ignore the negat
16、ive,Example 6,(a)On the same axis,graph Make sure x is in radians(b)Describe what is happening,Example 6:Solution,The graph of f(x)=sin x is moving right and left,Example 6:Notes,By subtracting a value d in the argument of f(x)=sin x,we move the functionLeft when d 0This is known as a horizontal tra
17、nslationThe value of d is added to the x-coordinate of every point on the graph of f(x)=sin x Dealing with horizontal translations is counter-intuitiveWhen d 0 the function looks like:f(x)=sin(x d)and we move it right,Commonly referred to as a phase shift,Example 7,(a)On the same axis,graph Make sur
18、e x is in radians(b)Describe what is happening,Example 7:Solution,The graph of f(x)=sin x is being stretched or compressed horizontally,Example 7:Notes,By multiplying the argument of f(x)=sin x by a value k weCompress the function horizontally when k 1Stretch the function horizontally when 0 k 1This
19、 is known as a horizontal dilationThe x-coordinate of every point on the graph of f(x)=sin x is multiplied by 1/kThe period is changed from 2 to:i.e.if k=2,the transformed function will have two periods in 2,Summary,Part 1,The graphs of sin,cos and tan have the following properties:,Summary,Part 2,W
20、e can transform the graphs of f(x)=sin x and f(x)=cos x in the following ways:Vertical translation f(x)=sin(x)+c or f(x)=cos(x)+c Move the axis of curve to y=cHorizontal translation f(x)=sin(x d)or f(x)=cos(x d)A phase shift of dVertical Dilation f(x)=asin(x)or f(x)=acos(x)Change amplitude to|a|(no negatives!)Horizontal Dilation f(x)=sin(kx)or f(x)=cos(kx)Change the period to,Practice Problems,P.258-260#1-11,15,19(not f)Note:Any graphs/sketches can be done using your TI-83 or the program“Graph”,
