For this assignment we will explore the effects of variables a and b on the graph of parametric curves x = acos(t) and y=bsin(t). This exploration may seem basic at first, but it is important to better comprehend trigonometric, parametric curves. Before we begin let's review the definition of a parametric curve: a parametric curve is generated by two functions where the two functions are defined by the pair (x, y). Take a look at how the input of our parametric equations is set up below. For the first curve we see x = 1cos(t) and y = 1sin(t). The variable t defines the range of our function. For this exploration we will see how the functions behave when t varies from 0 to 2π. Please make sure your browser is maximized before continuing.

So let's now take a look again at the first parametric curve where a and b equal +1. Does it look familiar? Yes, you got it! You are looking at what is called the unit circle. The unit circle is defined in Euclidean geometry as the circle with radius one centered about the origin . Sin(x) and cos(x) are defined on the unit circle for if the point (x, y) where x = cos(t) and y = sin(t) lies on the circle then the angle between the positive x-axis and the ray from (0, 0) and (x, y) will equal t. For our first parametric equation we see that at t = 0 we will be at the point (1, 0). You can reference the unit circle above to see where the point (x, y) will be for all the variations of t. But what happens when we multiply cos(t) and sin(t) by scalars a and b where a,b ∈ R?

By looking at the curves with functions multiplied by scalars a and b we see that the scalars dilate the curves by the amount of the scalar. The curves are either stretched or shrunk in size. So how are the scalars effecting our curves? If we go back to the definition of a parametric equation we recall that the input functions produce an output (x, y). Hence, when the function cos(t) is multiplied by the scalar a = 2 the curve cos(t) will be dilated by 2. For instance, when x = acos(π) is scaled by a = 2 we have the solution x = -2 instead of -1. Because our parametric equations x = acos(t) and y = bsin(t) form a variation of the unit circle depending on the values of a and b multiplying by a negative scalar will not effect the shape of the curve, but will change the direction of rotation. Note the congruency of the two curves above: x = 2cos(t) and y = 1sin(t) and x = -2cos(t) and y = -1sin(t).

We can step away from the unit circle and observe the curves cos(t) and sin(t) along the x-axis. These two curves will be effected by scalars in the same way as the parametric equations above. However, these two curves are viewed separately. They no longer form a curve of a parametric equation, but just the graph of the curves. Here, the functions are independent of one another.

 

Let's further our exploration and see what happens to our parametric curve when the scalars are moved inside the functions, i.e. x = cos(at) and y = sin(bt).

 

 

These curves may seem crazy and far beyond comprehension, but in reality they are behaving similarly to our first parametric curves. Let's first think about what we have done by moving our scalar a and b. Instead of the scalars dilating the functions as a whole the scalars are now dilating the parameter t. The parameter t is still varying from 0 to 2π. So the scalars a and b are dilating the value of t which then changes the angle between the positive x-axis and the ray from (0, 0) to the point (x, y). Recall t is the angle between the positive x-axis and the point (x, y). Notice also that the sign of the scalar does not effect the shape of the curve, like our first parametric curves.

In conclusion, we have explored the effects of scalars a and b on the functions x = cos(t) and y = sin(t) where t varies from 0 to 2π. We have discovered when the functions cos(t) and sin(t) are multiplied by real number scalars such that x = acos(t) and y = bsin(t) the value of the cos(t) and sin(t) are dilated such that the shape of the curve is a variation of the unit circle. We also discovered when the scalars are moved to scale the value of t our functions form new curves with varying angles, directions, and scales. The parametric equations x = cos(at) and y = sin(at) and x = acos(t) and y = bsin(t) show us the significance of scalars and their effects on parametric curves.

 

 

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