How does the magical black box produce values for sin(x)? Something as simple as a quadratic function can produce values for sin(x) with remarkable accuracy. Increase the degree of this polynomial slightly and the results are really impressive. This activity involves simple applications of the factor theorem, differential calculus and is a great launching point for the mean value theorem.
Understand how values for trigonometric functions are generated.
Sketch curves associated with simple polynomials; find stationary points, and local and global maxima and minima; and examine behaviour as x approaches positive / negative infinity.
Polynomial, degree (of polynomial), derivative, coefficient
About the Lesson
The purpose of this activity is to unveil some of the mystery behind the black box (calculator) that is able to generate trigonometric ratios with the click of a button. The unveiling uses simple polynomial functions and differential calculus. The activity begins with a simple quadratic that shares the x intercepts at x = 0 and x = pi, the quadratic is dilated so that it shares the same turning point in this domain, try it for yourself, the accuracy is really very impressive. A higher degree polynomial is then used to model sin(x), the flexibility of this new polynomial increases the accuracy quite significantly. When a calculus approach is used the resulting polynomial is able to produce sin(x) ratios with a four decimal accuracy, as good as the original ‘trig tables’.