This applet shows the Sine function and four approximations for the Sine function.[br][br]An early approximation for sine was from Bhaskara I is [br][math]\frac{16x\left(\pi-x\right)}{5\pi^2-4x\left(\pi-x\right)}[/math]. This function is fairly accurate for angles between [math]0\text{ and }\frac{\pi}{2}[/math].[br][br]Another approximation is a power series (MacLaurin or Taylor Series) which for the Sine function is [br][math]\sum_{n=1}^N\frac{\left(-1\right)^{n-1}x^{2n-1}}{\left(2n-1\right)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...[/math]. The number of terms can be increased to obtain any desired accuracy.[br]The third approximation is a product series that was used in the proof of [math]\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}[/math]. [br]Since the Sine function is zero at [br][math]x\in\left\{0,-\pi,\pi,-2\pi,2\pi,-3\pi,3\pi,...\right\}[/math] it can be approximated as [math]x\prod_{n=1}^N\left(1-\left(\frac{x}{n\pi}\right)^2\right)=x\left(1-\left(\frac{x}{\pi}\right)^2\right)\left(1-\left(\frac{x}{2\pi}\right)^2\right)\left(1-\left(\frac{x}{3\pi}\right)^2\right)...[/math]. This is the Root Products Approximation.[br][br][br]A fourth method is Polynomial Interpolation that go through a set of points. Two points define a line. For more points the polynomial can be defined as [math]\sum_{i=0}^n\left(\prod_{0\le j\le n,j\ne i}^{ }\frac{x-x_i}{x_i-x_j}\right)y_i[/math] using Lagrange Polynomials. Note, the value of the product is zero at all points [math]x=x_{j\ne i}[/math] and one when [math]x=x_i[/math]. Here the points are marked with small squares and are at well known Sine angles, [math]0,\pm\frac{\pi}{6},\pm\frac{\pi}{4},\pm\frac{\pi}{3},\pm\frac{\pi}{2},\cdots[/math] where sine is [math]0,\pm\frac{1}{2},\pm\frac{1}{\sqrt{2}},\pm\frac{2}{\sqrt{3}},\pm1[/math]. This is labeled Lebesgue.[br][br][br]Compare the different approximations as extra terms are added.[br][br]How many terms are required for the MacLaurin Series to achieve grater accuracy than the other approximations?[br][br]Where do the different approximations provide more accurate values?