As x varies, the value of the cosine varies with the variation in the length of OQ. At the midpoint of the second half of the period (that is, at x = 3π/2), the height of the sine wave is y = −1 draw that dot.As shown in the image above, we note that cos x = OQ/OP = OQ/1 = OQ. At the midpoint of the first half of the period (that is, at x = π/2), the height of the sine wave is y = 1 draw that dot. In the middle of the period, for x = π, the sine wave is again crossing the x-axis. At either end of the standard period (that is, where x = 0 and x = 2π), the sine wave is at y = 0 (that is, it's on or crossing the x-axis). Plot these five "interesting" points, and then fill in the curve.Ĭorrespondingly, when you do your sine graphs, use the five useful points. The midpoints of each half of the standard period are at x = π/2 and x = 3π/2 at these points, the cosine wave is at y = 0 (that is, it's on or crossing the x-axis). In the middle of the period, for x = π, the cosine wave is at y = −1. You might want to when you're a beginner, but you will quickly notice that some points of these waves are more useful (interesting, regular) than nearly all of the others.įor instance, at either end of the standard period (that is, where x = 0 and x = 2π), the cosine wave is at height y = 1. When you graph, it isn't necessary to plot loads of points. (The amplitude would also be 3 for the function h( x) = −3cos( x) the only difference would be that this wave would be upside-down from the regular cosine wave.) How can I easily graph sines and cosines? In this case, the amplitude of the function would be 3. For instance, if you were given g( x) = 3cos( x), the multiplier " 3" will multiply all of the cosine's values, so that the curve will vary between −3 and +3. If the sine or cosine function is transformed by multiplication, then this will change the amplitude. So even if you were given the function f( x) = sin( x) + 4, so the wave was centered four units above the x axis, the wave would go no higher than 5 and no lower than 3 their amplitudes would still be 1. But even the pushed-up or pulled-down sines and cosines will still wave up and down a fixed distance above and below their midlines. But sines and cosines can be translated up or down by adding or subtracting some number to the function. Note that the sine and cosine curves go one unit above and below their midlines here, the midline happens to be the x-axis. This value of " 1" is called the "amplitude" of the waves. The sine and cosine functions each vary in height, as their waves go up and down, between the y-values of −1 and +1. When you hand-draw graphs, you should instead always use the exact values: π, 2π, π/2, etc.)
Cosine graph software#
(Note: In the graphs above, my horizontal axes are labelled with decimal approximations of π because that's all my equation-grapher software can handle. The Sine Waveįrom the above graph, which shows the sine function from −3π to +5π, you can probably guess why the graph of the sine function is called the sine "wave": the circle's angles repeat themselves with every revolution of the unit circle, so the sine's values repeat themselves with every length of 2π, and the resulting curve is a wave, forever repeating the same up-and-down wave. Therefore, by unwrapping the sine from the unit circle and rolling it out sideways, we have been able create a function: the sine function, designated as " sin()" (or possibly just on your calculator). In other words, we would have this graph:Īs you can see from the graphic, each input value (each angle measure, which is also an x-value) corresponds with (spits out, results in) a single output value (a sine value, which is also a y-value).
If, instead of starting over again at zero for every revolution on the unit circle, we'd counted up higher angle measuress each time we re-entered the first quadrant in the unit-circle part of the sine-value animation, then the graph on the right would have continued, up and down, over and over again, past 2π and onward to the right. If the green angle-line in the unit-circle part of the sine-value animation above had gone backwards (that is, in reverse) counting into negative angle measures, the graph on the right would then have extended back to the left of zero. But we don't have to restrict ourselves to only this interval of angle values we are allowed to keep counting upward, past 2π, and backwards, before 0π, rather than resetting each rotation. We typically think of the angles as going from 0π up to (but not quite including) 2π, with the angle-measure resetting each time we re-enter the first quadrant. Now let's think a bit more about the unit circle.
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