Activity 5.3.4.
We understand the theoretical rule behind the function \(f(t) = \sin(t)\text{:}\) given an angle \(t\) in radians, \(\sin(t)\) measures the value of the \(y\)-coordinate of the corresponding point on the unit circle. For special values of \(t\text{,}\) we have determined the exact value of \(\sin(t)\text{.}\) For example, \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\text{.}\) But note that we don’t have a formula for \(\sin(t)\text{.}\) Instead, we use a button on our calculator or command on our computer to find values like “\(\sin(1.35)\text{.}\)” It turns out that a combination of calculus and polynomial functions explains how computers determine values of the sine function.
At http://gvsu.edu/s/0zA, you’ll find a Desmos worksheet that has the sine function already defined, along with a sequence of polynomials labeled \(T_1(x)\text{,}\) \(T_3(x)\text{,}\) \(T_5(x)\text{,}\) \(T_7(x)\text{,}\) \(\ldots\text{.}\) You can see these functions’ graphs by clicking on their respective icons.
(a)
(b)
(c)
(d)
What overall trend do you observe? How good is the approximation generated by \(T_{19}(x)\text{?}\)
(e)
In a new Desmos worksheet, plot the function \(y = \cos(x)\) along with the following functions: \(P_2(x) = 1 - \frac{x^2}{2!}\) and \(P_4(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!}\text{.}\) Based on the patterns with the coefficients in the polynomials approximating \(\sin(x)\) and the polynomials \(P_2\) and \(P_4\) here, conjecture formulas for \(P_6\text{,}\) \(P_8\text{,}\) and \(P_{18}\) and plot them. How well can we approximate \(y = \cos(x)\) using polynomials?
