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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 oimage12.pngimage12.png f, and how that relates to the graph of the sine curve. In particular, it would be investigated how the different variables (image38.pngimage38.png 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 image73.pngimage73.png and image35.pngimage35.png to be 0, which would mean that the equation image115.pngimage115.png takes the form of: image13.pngimage13.png. It can be seen from Graph 1.1 that when image15.pngimage15.png is 1 what graph you get (the red graph) and when image15.pngimage15.png is allowed to be 2 what the graph looks like (the blue graph). From the graph below it can be seemn that increasing image15.pngimage15.png stretches the graph by the factor of change in image15.pngimage15.png. In simple words, the graph of image39.pngimage39.png would be twice the height of image25.pngimage25.png as is clearly seen from Graph 1.1.

When I change the value for image15.pngimage15.png, all the value of the sine curve get multiplied by that value of image15.pngimage15.png, which is 2 in this case. By multiplying all the values of the curve by image15.pngimage15.png 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 (image25.pngimage25.png) it is common knowledge that the relative minima and maxima are -1 and 1 respectively, however when the image15.pngimage15.png is changed the original minima and maxima are also multiplied by image15.pngimage15.png and therefore the new minima and maxima would become image62.pngimage62.png and image15.pngimage15.png respectively. The rest of the graph also gets stretched by image15.pngimage15.png as this is the number that the whole curve is being multiplied by. image47.pngimage00.png

        This time when we change the value of image15.pngimage15.png

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Middle

In general it is observed that in the equation image31.pngimage31.png the period of the curve is given by image32.pngimage32.png. By changing the value of image14.pngimage14.png you can change the period of the graph.  When you increase the image14.pngimage14.png the period of the graph decreases and when you decrease the image14.pngimage14.png 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. image06.png

        When you make image34.pngimage34.png and image35.pngimage35.png 0 you get the equation image36.pngimage36.png. In this equation, image37.pngimage37.png corresponds to the horizontal translation of the graph. When image37.pngimage37.png is a positive number the translation is a horizontal translation to the right, whereas when the value of image37.pngimage37.png is negative the graph translates to the left by image37.pngimage37.png units as illustrated by Graph 3.1. image33.png

When you change the value of image37.pngimage37.png you subtract the original value of image16.pngimage16.png by image37.pngimage37.png, and therefore the image16.pngimage16.png value of the new curve would be increased by image37.pngimage37.png to match the image23.pngimage23.png value. For example, when image40.pngimage40.png, where image37.pngimage37.png is 0, image41.pngimage41.png. However when there is a value for image37.pngimage37.png,image42.pngimage42.png cannot be 0 unless image43.pngimage43.png in which case image16.pngimage16.png must be equal to image44.pngimage44.png. This would translate the point and the rest of the graph by image37.pngimage37.png 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 image37.pngimage37.png is a negative number, the equation becomes image45.pngimage45.png and this causes the graph to translate image37.pngimage37.png units to the left.  image07.png

        From Graph 3.2 it can be seen that the value of image37.pngimage37.png 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 image37.pngimage37.png

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Conclusion

image103.pngimage103.png looks just like the curve of image25.pngimage25.png and therefore could be a translation. Just like the sine curve, the amplitude (image15.pngimage15.png) of the cosine curve is 1, whereas, the period (image14.pngimage14.png) of the cosine curve is image104.pngimage104.png (just like sine curve), meaning that the image14.pngimage14.png 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 image26.pngimage26.png units, meaning that the value of image105.pngimage105.png and the value of image63.pngimage63.png. When we plug the numbers into the equation of the sine curve image31.pngimage31.png the equation that we get is the following: image106.pngimage106.png. Since this equation is for the cosine curve we can say that image107.pngimage107.png. However, there is another way to express this translation. It can be said that the graph of image25.pngimage25.png was flipped with respect to the image16.pngimage16.png axis and then translated horizontally to the right by image26.pngimage26.png units. When this is the case image15.pngimage15.png becomesimage108.pngimage108.png, image14.pngimage14.png and image35.pngimage35.png remain 0 and image37.pngimage37.png changes from image109.pngimage109.png to image26.pngimage26.png. The equation for this new translation would look like image110.pngimage110.png and is something else which could be used for calculations in trigonometry. image03.png

        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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