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Draw the graphs of y1 = Sin x and y2 = Cos x for ranging from 0 to 2π, and then by the addition of ordinates, obtain the graph of y = Sin x + Cos x. What is the period of Sin x + Cos x and also the amplitude?
Solution:
The first thing that we have to do is to create the table so that it is easier for us to compute their values of y. By the way, all the units are unitless and so their angles must be expressed in radians. You can use calculator so that you can get the accurate value of y.
Note: The conversion of degrees to radians is given by the formula
Next, use your scientific calculator to compute for their values of trigonometric functions as follows
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Since we have now the values of x and y, we can start now the graphing of the given trigonometric functions. First, let's start with the Sine function. As you notice that the trend of the graph is a wave because of the periodically repeating the values of y but the highest and the lowest values are always 1 and -1.
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For the Cosine function, the trend of the graph is almost the same but it didn't pass through the origin and the highest and the lowest values are always 1 and -1 also.
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To draw the Sin x + Cos x function, you need a compass or divider for this. Using a divider, measure the height or the value of y of the cosine graph from the x-axis and then transfer it or plot the point to the sine function from the curve proper. Connect all the plotted points and the trend of the graph will be like this.
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| Photo by Math Principles in Everyday Life |
If you don't have a compass or a divider, you can use the values of y = Sin x + Cos x from the table to plot the points directly and then connect all the points. As you noticed that the resulted graph is also a repeated wave because of the periodically repeating the values of y also.
From the trend of a graph and the table, the amplitude or the height of the curve is 1.4142 (+ and -) which is also equal to √2.
To compute for the period, you need the graph proper to see the trend of a curve. The trend of a curve that we need is a one complete wave or one cycle which is started from y = 0 and then ends at y = 0 that passed the minimum and maximum amplitudes. By looking at the graph, the approximate value of a period is 2.3 - (- 3.9) = 2.3 + 3.9 = 6.2 radians.
There's another way to compute for the period. You can also use the table and look for the value of y = Sin x + Cos x. As you can see that if y = 0 then x = 2.3562 radians and if y = 0 then x = 5.4978 radians. Therefore, the value of a period is 2(5.4978 - 2.3562) = 6.2852 radians which also equal to 360°.