Active Prelude to Calculus

In Section 2.1, we saw how tracking the height of a point that is traversing a cirle generates a periodic function, such as in Figure 2.1.7. Then, in Section 2.2, we identified a collection of \(16\) special points on the unit circle, as seen in Figure 2.3.1.

You can also use the Desmos file at http://gvsu.edu/s/0xt to review and study the special points on the unit circle.

The circular function that tracks the height of a point on the unit circle traversing counterclockwise from \((1,0)\) as a function of the corresponding central angle (in radians) is one of the most important functions in mathematics. As such, we give the function a name: the sine function.

Because of the correspondence between radian angle measure and distance traversed on the unit circle, we can also think of \(\sin(t)\) as identifying the \(y\)-coordinate of the point after it has traveled \(t\) units counterclockwise along the circle from \((1,0)\text{.}\) Note particularly that we can consider the sine of negative inputs: for instance, \(\sin(-\frac{\pi}{2}) = -1\text{.}\)

Based on our earlier work with the unit circle, we know many different exact values of the sine function, and summarize these in Table 2.3.4.

Moreover, if we now plot these points in the usual way, as we did in Preview Activity 2.3.1, we get the familiar circular wave function that comes from tracking the height of a point traversing a circle. We often call the graph in Figure 2.3.5 the sine wave.

Given any central angle of radian measure \(t\) in the unit circle with one side passing through the point \((1,0)\text{,}\) the angle generates a unique point \((a,b)\) that lies on the circle. Just as we can view the \(y\)-coordinate as a function of \(t\text{,}\) the \(x\)-coordinate is likewise a function of \(t\text{.}\) We therefore make the following definition.

Again because of the correspondence between the radian measure of an angle and arc length along the unit circle, we can view the value of \(\cos(t)\) as tracking the \(x\)-coordinate of a point traversing the unit circle clockwise a distance of \(t\) units along the circle from \((1,0)\text{.}\) We now use the data and information we have developed about the unit circle to build a table of values of \(\cos(t)\) as well as a graph of the curve it generates.

As we work with the sine and cosine functions, it’s always helpful to remember their definitions in terms of the unit circle and the motion of a point traversing the circle. At http://gvsu.edu/s/0xe you can explore and investigate a helpful Desmos animation that shows how this motion around the circle generates each of the respective graphs.

Because the sine function results from tracking the \(y\)-coordinate of a point traversing the unit circle and the cosine function from the \(x\)-coordinate, the two functions have several shared properties of circular functions.

It is also insightful to juxtapose the sine and cosine functions’ graphs on the same coordinate axes. When we do, as seen in Figure 2.3.8, we see that the curves can be viewed as horizontal translations of one another.

In light of the definitions of the sine and cosine functions, we can now view any point \((x,y)\) on the unit circle as being of the form \((\cos(t),\sin(t))\text{,}\) where \(t\) is the measure of the angle whose vertices are \((1,0)\text{,}\) \((0,0)\text{,}\) and \((x,y)\text{.}\) Note particularly that since \(x^2 + y^2 = 1\text{,}\) it is also true that \(\cos^2(t) + \sin^2(t) = 1\text{.}\) We call this fact the Fundamental Trigonometric Identity.

There are additional trends and patterns in the two functions’ graphs that we explore further in the following activity.

We have established that we know the exact value of \(\sin(t)\) and \(\cos(t)\) for any of the \(t\)-values in Table 2.3.4, as well as for any such \(t \pm 2j\pi\text{,}\) where \(j\) is a whole number, due to the periodicity of the functions. But what if we want to know \(\sin(1.35)\) or \(\cos(\frac{\pi}{5})\) or values for other inputs not in the table?

Any standard computing device such as a scientific calculator, Desmos, Geogebra, or WolframAlpha, has the ability to evaluate the sine and cosine functions at any input we desire. Because the input is viewed as an angle, each computing device has the option to consider the angle in radians or degrees. It is always essential that you are sure which type of input your device is expecting. Our computational device of choice is Desmos. In Desmos, you can change the input type between radians and degrees by clicking the wrench icon in the upper right and choosing the desired units. Radian measure is the default.

It takes substantial and sophisticated mathematics to enable a computational device to evaluate the sine and cosine functions at any value we want; the algorithms involve an idea from calculus known as an infinite series. While your computational device is powerful, it’s both helpful and important to understand the meaning of these values on the unit circle and to remember the special points for which we know the outputs of the sine and cosine functions exactly.