Breakdown+of+a+Sine+wave

=Modelling a Sine Wave=

Before we introduce the unit circle, we must first understand the basics used to derive the unit circle. in a circle, the line from the origin to the outer point has a radius of 1. This is indicated by the coordinates on the picture below (0,1), (1, 0), (-1,0) and (0, -1). With this, we can then derive the other points in the unit circle. In particular, the angles that are labelled on the unit circle below. These angles occur every 30, 45 and 60 degrees of the unit circle and can be represented in fraction form, as shown outside the unit circle. they can also be described in terms of "radian" or in terms of pi.

As you can see, the numbers inside of the circle and outside the numerical degrees describe each rotation of the circle, not in terms of degree, but in terms of radians ( pi/ ). A radian is actually the length of the radius described in terms of the outer edge of the circle. Or if you wanted to imagine, you take the length of the radius, and bend it along the outside of the circle, it would equal to 1 radian. The radian of one circle is always constant no matter how big or small the circle is. In terms of degrees, this would equal to 57.2958. However, using decimals when calculating other formulas is very impractical, so from this point onwards, we will describe the unit circle in terms of radian. Knowing that, we can then calculate that one "pi" is equal to 180 degrees. furthermore, 2pi = 360 degrees. Furthermore, taking this knowledge, we can then calculate every single point that occurs on the unit circle in terms of radians.



Taking all of the knowledge of the unit cirlce, we can then extrapolate the data from the unit circle, and graph the information. on the Y-axis, we will have the amplitude of the sine wave. In simpler terms, this would equal to the deviation from the centre of the wave (in this particular graph, this would be shown on the X-axis). on the X-axis, we will have measurements in terms of radians. In this picture, we can see that one cycle is equal to 2pi. However, in different graphs, this can be changed to many different cycle lengths and amplitudes.

Now that we know the basics of a sine graph, we can then move on to the manipulation of a sine wave. It is quite simple really. To graph a sine wave, one must follow this equation:

F(x) = (a)sinb+k

In this equation, we can manipulate the variables a, b, x, h and k to change the shape of the sin wave.

a: describes the deviation from the origin with the form |a|. For example, the graph above has a deviation of 1.

b: describes the horizontal deviation from the origin. For example, if the b value of a certain graph equated to -1, then the center of the graph would start at -1 (the graph would start upwards from the point -1,0). In the graph above, the b value is 0. Also, from the b value, we can find the period length of a sin graph by plugging in the value of b into the form (2pi/b) and evaluating.

k: describes the vertical deviation of the origin of the sin graph itself. For example, the graph has a k value of 0.