where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians =


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1 Introduction to Modeling Sine and Cosine Functions The general form of a sine or cosine function is given by: f (x) = asin (bx + c) + d and f(x) = acos(bx + c) + d where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians = o 180, the angle measure of a half circle.) Their grahs look like a wave, with one wave segment reeated continuously. The simlest versions of these functions are f ( x) = sin( x) and f ( x) = cos( x) in the case where a = b = 1 and c = d = 0. f ( x) = sin( x) f ( x) = cos( x) Notice that the grahs look very similar, excet the cosine function is moved to the left by an amount of π. Indeed, every cosine function is a shifted sine function, and therefore, we will concentrate on sine functions only from now on. Activity For the functions given below, identify the values of the constants a, b, c, and d. 1) f ( x) = sin( x) f ( x) = sin( x)
2 3.6 Silvia Heubach ) f ( x) = sin( x) f ( x) = sin( x) 3) f ( x) = sin( x) f ( x) = sin( x) 1 Looking at the airs of grahs for examles 1), ), and 3), notice that only one of the constants is different. This means that any change in the grah is caused by that articular constant. Let s consider each air in detail: 1) Here the constant a changed from 1 to. The grah now reaches twice as high and twice as low. This roerty is characterized by the amlitude. The amlitude of a sine/cosine function is given by a. To read off the amlitude from the data, comute the difference between the maximal and minimal outut values (to measure vertical distance) and divide by. In the examle given below, the amlitude is comuted as a = ( ( )) =. 3 1 high amlitude low
3 Introduction to Modeling Thus, the amlitude gives half the amount of the overall variation in outut values and measures how much the extreme high and low values differ from the average value. ) In this case, the constant b has changed from 1 to. If we comare the grahs for inut values from 0 to π, on the right grah we see two reetitions of the basic segment (= one full wave). Therefore, b indicates the frequency of a sine/cosine curve, i.e., the number of reeated segments in an interval of length π. In alications, we are often interested in a closely related value, the eriod, which gives the length of the reeated segment or cycle. Since b equals the number of reeated segments in an interval of length π, we have 1443 = π # of segments length of segment This formula rovides an easy way to comute the eriod as we can easily read off the value of b from the functional exression. If we only have data, then we can read off the eriod as follows: Select two consecutive oints at the same outut level where the function is either both increasing or decreasing. (Simle oints to consider are two consecutive high or low oints.) The horizontal distance (= difference in inut values) between these oints gives the eriod. In the grah below, the eriod is π ) Here, the only constant that has changed is d, from 0 to 1. The effect in the grah is a vertical shift downward by 1 unit. Thus, d gives the vertical shift; negative values corresond to a downward shift while ositive values corresond to an uward shift. A second interretation of the constant d is the average or middle value for the sine function. To find the value of d from the grah, we add the maximal and minimal outut values, then divide by. From the second grah of 3) in Activity 3.6.1, we see that the biggest outut value is 0 and the lowest one is. Thus, d = ( 0 + ( )) = 1. The remaining constant in the formula for the sine function is c. This constant influences the horizontal shift of the sine function, but can not always be read off directly from the grah.
4 3.64 Silvia Heubach However, in ractical alications, the values of the other three constants are more imortant. Furthermore, the horizontal shift is not uniquely determined because of the reetitive nature of the sine function. Here is an examle on how to read off the relevant arameters for the data given below: x f ( x) The grah of the data shows a wavelike shae. If we assume for the moment that this behavior will continue, we can use the grah and the table to determine the values of the constants a, b and d of a sine function. The amlitude a gives the amount by which the extreme values differ from the average value. We comute a as half the difference between maximal and minimal outut value. From the table we see that the largest outut value (in the f ( x) column) is 1, and the smallest outut value is 5. Thus, the amlitude is a = ( 1 ( 5) ) ( 1+ 5) To comute the value of the frequency b, we read off the eriod (= length of reeated segment) and then use the formula connecting frequency and eriod. Recall that we are looking for oints at which the outut values are the same (or aroximately the same), and the behavior of the function is identical (either increasing or decreasing values from left to right). Thus, even though the outut values are the same for the two oints (0.785, 0.11) and (.356, 0.11), the behavior of the function near those two oints is different. For the first oint, the function increases, while for the second oint, the function decreases. The only oints that can be used are (0, ) and (6.83, ). The eriod is the horizontal distance between these oints, comuted as the difference in inut values. Thus, we have: = = 3
5 Introduction to Modeling eriod = = 6.83 and therefore, (since b = π) π b = = = Last, but not least, we comute d, the average value (or vertical shift). To comute d we add the maximal and minimal outut values, then divide by : ( 1+ ( 5) ) 4 d = = =. Remark: Be careful not to confuse the comutation of amlitude and average value. Both involve maximal and minimal outut values, but for the amlitude the values are subtracted, whereas for the average value they are added, before being divided by. Activity 3.6. For the given data, determine from the grah whether the data follows a sine function (giving reasons for your answer). If a sine function is likely, comute the amlitude, eriod, and vertical shift for the function. Indicate which oints you have selected to comute these three quantities and use the table of values to do the necessary comutations. x f ( x)
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