Trigonometric functions - Mathematics resources

1 Trigonometricfunctionsmc-TY-trig-2009-1 The sine, cosine and tangent of an angle are all defined in terms of trigonometry, but they canalso be expressed as functions . In this unit we examine thesefunctions and their graphs. Wealso see how to restrict the domain of each function in order to define an inverse order to master the techniques explained here it is vital that you undertake plenty of practiceexercises so that they become second reading this text, and/or viewing the video tutorial on this topic, you should be able to: specify the domain and the range of the three Trigonometric functionsf(x) = sinx,f(x) =cosxandf(x) = tanx, understand the difference between each function expressed in degrees and the correspondingfunction expressed in radians, express the periodicity of each function in either degrees or radians, specify a suitable restriction for the domain of each function so that an inverse functioncan be defined, find the appropriate value ofx(in either degrees or radians) when given a value ofsinx, sine functionf(x) = cosine functionf(x) = tangent functionf(x) = mathcentre 20091.

2 IntroductionIn this unit we shall use information about the Trigonometric ratios sine, cosine and tangent todefine functionsf(x) = sinx,f(x) = cosxandf(x) = The sine functionf(x) = sinxWe shall start with the sine function,f(x) = sinx. This function can be defined for any numberxusing a diagram like x1We take a circle with centre at the origin, and with radius 1. We then draw a line from theorigin, atxdegrees from the horizontal axis, until it meets the circle,so that the line has length1. We then look at the vertical axis coordinate of the point where the line and the circle meet,to find the value information from this picture can also be used to see how changingxaffects the value ofsinx. We can use a table of values to plot selected points betweenx= 0 andx= 360 , anddraw a smooth curve between them.

3 We can then extend the graphto the right and to the left,because we know that the graph repeats 45 90 135 180 225 270 315 360 1 10360 720 360 f(x)xf(x) = sin mathcentre 2009 Whenx= 0,sinx= 0. As we increasexto90 ,sinxincreases to 1. As we increasexfurther,sinxdecreases. It becomes zero whenx= 180 . It then continues to decrease, and becomes 1whenxis270 . After thatsinxincreases and becomes zero again whenxreaches360 . Wehave now come back to where we started on the circle, so as we increasexfurther the can also use this picture to see what happens whenxis less than zero. If we decreasexfromzero,sinxdecreases. It becomes 1whenx= 90 . Then it becomes zero atx= 180 , and1 atx= 270 . It then decreases and becomes zero whenx= 360.

4 This cycle is repeated ifwe this picture we can see that, whatever value we pick forx, the value ofsinxmust alwaysbe between 1and 1. So the domain off(x) = sinxcontains all the real numbers, but therange is 1 sinx 1. We can also see that the function repeats itself every360 . We cansay thatsinx= sin(x+ 360 ). We say the function is periodic, with periodicity360 .Sometimes we will want to work in radians instead of we havesinxin radians, it isusually very different fromsinxin degrees. For examplesin 90 = 1but in radianssin(90)isabout We can use a table of values like the one we had before to plot a graph ofsinxinradians. As2 radians is the same as360 the graph will be very similar to the graph forxindegrees, but now the labels on the axes have 4 23 4 5 43 27 42 1 102 4 2 f(x)xf(x) = sin mathcentre 2009To compare the two graphs, we can keep the same scale on thex-axis and plot both (x)x60x in degreesx in radiansWithxin degrees, the functionf(x) = sinxhas not reached 1 by the right-hand side of thegraph, but withxin radians the function has oscillated several times.

5 So these are quite , instead of finding the sine of an angle, we want to work backwards. We want tofind an angle whose sine is, say,34. So we want to define a new function to give us the inversesine of the number. We want to find a function such thatf 1(x) =ywheneverf(y) =x. Inour case, we wantsinx=34, so that we shall want to havesin 1(34) = this might seem to be a problem at first because, if we look back at our graph, we see thatthere are lots of angles withsinx= 10360 720 360 34f(x)xf(x) = sin xWe cannot define a function to tell us what the inverse sine of34should be if there is a choice ofvalues forf 1(x). To get around this problem, we need to restrict the domain ofour functionf(x) = sinxso that we have only a part of the graph that gives us one angle for each sine happens if we cut our domain down to 90 x 90 , or x if we work mathcentre 20091 1090 90 34f(x)xf(x) = sin xWe say that our functionf(x) = sinxhas domain 90 x 90 and that it has an inverse,f 1(x) = sin 1x.

6 This inverse function is also written asarcsinx. So, if the anglexlies in therange 90 x 90 andsinx=34, we sayx= sin 1(34). You can use your calculator to workout inverse PointThe functionf(x) = sinxhas all real numbers in its domain, but its range is 1 sinx values of the sine function are different, depending on whether the angle is in degrees orradians. The function is periodic with periodicity 360 degrees or 2 can define an inverse function, denotedf(x) = sin 1xorf(x) = arcsinx, by restricting thedomain of the sine The cosine functionf(x) = cosxWe shall now look at the cosine function,f(x) = cosx. This function can be defined for anynumberxusing a diagram like mathcentre 2009We take a circle diagram similar to the one we used for the sinefunction.

7 But now we look atthe horizontal axis coordinate of the point where the line and the circle meet, to find the information from this picture can also be used to see how changingxaffects the value ofcosx. We can use a table of values to plot selected points betweenx= 0 andx= 360 , anddraw a smooth curve between them. We can then extend the graphto the right and to the left,because we know that the graph repeats 45 90 135 180 225 270 315 360 1 10360 720 360 f(x)xf(x) = cos xAgain you can see thatcosxmust lie between 1and 1. This function also has periodicity360 ,or2 if we work in withsinx, we should like to define an inverse function to tell us the angle having a cosine of,say,12. Unless we restrict the domain ofcosx, there will be many angles that could becos we defined the inverse sine function, we restricted the domain ofsinxto 90 x 90.

8 Let us see what happens if we do this 1090 90 f(x)xf(x) = cos mathcentre 2009In this case we still have two angles for some values of the cosine function. Also, if we look atnegative values then then there are no angles at all. So we need to choose a different domain. Agood choice is0 x 180 , because this has only one angle giving each possible if we restrict the domain off(x) = cosxin this way, we can definecos 10180 f(x)xf(x) = cos x12So now we say that our functionf(x) = cosxhas domain0 x 180 and that it has aninverse,f 1(x) = cos 1x. This inverse function is also written asarccosx. So, if the anglexlies in the range0 x 180 andcosx=12, we sayx= cos 1(12).Key PointThe functionf(x) = cosxhas all real numbers in its domain, but its range is 1 cosx values of the cosine function are different, depending onwhether the angle is in degrees orradians.

9 The function is periodic with periodicity 360 degrees or 2 can define an inverse function, denotedf(x) = cos 1xorf(x) = arccosx, by restrictingthe domain of the cosine function to0 x 180 or0 x .4. The tangent functionf(x) = tanxFinally we deal withtanx, which is justsinx/cosx. We can use a table of values to plot selectedpoints betweenx= 0 andx= 360 , as before. But now we can use the values forsinxandcosxthat we have already mathcentre 2009x0 45 90 135 180 225 270 315 360 1 1 * 101* 10 You can see from the table that there are some values ofxfor whichcosx= 0, and sotanxisnot defined for these values ofx. These are at90 ,270 , and also other values differing fromthem by multiples of180.

10 1 10360 720 360 f(x)xf(x) = tan x 22 Notice that, when we get tox= 360 , the graphs ofsinxandcosxrepeat themselves. Astanxdepends on onlysinxandcosx, the graph oftanxmust also repeat itself. Buttanxrepeatsitself more often thansinxandcosx. It repeats itself every180 . Sotanxhas a periodicity of180 , or if you work in radians. Notice also that, unlikesinxandcosx, the functiontanxdoesnot have to lie between 1and 1. In facttanxcan take any can also define an inverse tangent function, and to do this we must restrict the domainoff(x) = tanx. In fact we need to think twice as hard this time because, as well as makingsure that we get only one angle giving each tangent value, we must also avoid trying to definef(x) = tanxover a region where there is a zero cosine value.