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# Mathematics Higher Level Internal Assessment Investigating the Sin Curve

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Introduction

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Investigating the Sine Curve

This report investigates the sine curve in the form o f, and how that relates to the graph of the sine curve. In particular, it would be investigated how the different variables ( effect the way that the graph is drawn and then seeing if the rule can be generalized to apply to any form of the equation.

The first thing to do would be to allow  and  to be 0, which would mean that the equation  takes the form of: . It can be seen from Graph 1.1 that when  is 1 what graph you get (the red graph) and when  is allowed to be 2 what the graph looks like (the blue graph). From the graph below it can be seemn that increasing  stretches the graph by the factor of change in . In simple words, the graph of  would be twice the height of  as is clearly seen from Graph 1.1.

When I change the value for , all the value of the sine curve get multiplied by that value of , which is 2 in this case. By multiplying all the values of the curve by  the height of each point in the curve increases while there is no change in the position of the graph in the x-axis. For the original sine graph () it is common knowledge that the relative minima and maxima are -1 and 1 respectively, however when the  is changed the original minima and maxima are also multiplied by  and therefore the new minima and maxima would become  and  respectively. The rest of the graph also gets stretched by  as this is the number that the whole curve is being multiplied by.

This time when we change the value of

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Middle

In general it is observed that in the equation  the period of the curve is given by . By changing the value of  you can change the period of the graph.  When you increase the  the period of the graph decreases and when you decrease the  the period of the graph increases. This means that the curve only stretches in the horizontal direction while there is no stretch in the vertical direction of the curve.

When you make  and  0 you get the equation . In this equation,  corresponds to the horizontal translation of the graph. When  is a positive number the translation is a horizontal translation to the right, whereas when the value of  is negative the graph translates to the left by  units as illustrated by Graph 3.1.

When you change the value of  you subtract the original value of  by , and therefore the  value of the new curve would be increased by  to match the  value. For example, when , where  is 0, . However when there is a value for , cannot be 0 unless  in which case  must be equal to . This would translate the point and the rest of the graph by  units to the right. This would only cause a translation meaning that there is no change in the amplitude or the period of the curve. From Graph 3.1, it can also be seen that that when  is a negative number, the equation becomes  and this causes the graph to translate  units to the left.

From Graph 3.2 it can be seen that the value of  can be any number. It can either be a fraction, a whole number or even an irrational number. The same concept follows for fractions and irrational numbers as well, where the value of

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Conclusion

looks just like the curve of  and therefore could be a translation. Just like the sine curve, the amplitude () of the cosine curve is 1, whereas, the period () of the cosine curve is  (just like sine curve), meaning that the  value of the cosine curve would be 1. The only difference in the sine and cosine curve seems to be that the cosine curve has been translated to the left hand side by  units, meaning that the value of  and the value of . When we plug the numbers into the equation of the sine curve  the equation that we get is the following: . Since this equation is for the cosine curve we can say that . However, there is another way to express this translation. It can be said that the graph of  was flipped with respect to the  axis and then translated horizontally to the right by  units. When this is the case  becomes,  and  remain 0 and  changes from  to . The equation for this new translation would look like  and is something else which could be used for calculations in trigonometry.

In conclusion, it was observed that there is a strong relationship between the variables of an equation and how the graph looks. In terms of the sine curve, the variables represent the amplitude, the period and the translations (horizontal and vertical) of the curve. This allowed us to come up with a graph without having to plot it, and when we plotted the graph our guess was shown to be correct indicating that the idea of the variables was correct. Near the end of the investigation, we used these variables and the sine curve to come up with a relationship between the sine and cosine curve which would then help us in future trigonometry calculations. Just like that, we can deduce different relationships between different sorts of sine, and cosine functions to help us in future calculations.

Greg McLean

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