Roots Of Trigonometric Functions example essay topic

Finding roots of a function is often a task which faces mathematicians. For simple functions, such as linear ones, the task is simple. When functions become more complex, such as with cubic and quadratic functions, mathematicians call upon more convoluted methods of finding roots. For many functions, there exist formulas which allow us to find roots. The most common such formula is, perhaps, the quadratic formula.

When functions reach a degree of five and higher, a convenient, root-finding formula ceases to exist. Newton's method is a tool used to find the roots of nearly any equation. Unlike the cubic and quadratic equations, Newton's method - more accurately, the Newton-Raphson Method - can help to find roots of nearly any type of function, including all polynomial functions. Newton's method use derivative calculus to find the roots of a function or relation by first taking an approximation and then improving the accuracy of that approximation until the root is found. The idea behind the method is as follows. Given a point, P (Xn, Yn), on a curve, a line tangent to the curve at P crosses the X axis at a point whose X coordinate is closer to the root than Xn.

This X coordinate, we will call Xn+1. Repeating this process using Xn+1 in place of Xn will return a new Xn+1 which will be closer to the root. Eventually, our Xn will equal our Xn+1. When this is the case, we have found a root of the equation. This method may be unnecessarily complex when we are solving a quadratic or cubic equation.

However, the Newton-Raphson Method compensates for its complexity in its breadth. The following examples show the versatility of the Newton Raphson Method. Example 1 is a simple quadratic function. The most practical approach to finding the roots of this equation would be to use the quadratic equation or to factor the polynomial.

However, the Newton-Raphson method still works and allows us to find the roots of the equation. The initial number, Xn, 3, is a relatively poor approximation. The choice of 3 illustrates that the initial guess can be any number. However, as the initial approximation worsens, the calculation becomes more laborious. Example 2 demonstrates one of the advantages to Newton's method. Function 2 is a Quintic function.

Mathematician, Niels Henrik A bels proved that there exists no convenient equation, such as the cubic equation, which can help us find the function's roots. Thus, we are left without the crutch that helps us solve quadratics and cubic's. Using Newton's method is inevitable in this case because other, less mechanical and structured methods, such as factoring, would leave us with inconclusive results after long calculations. Despite the fact that, again, our approximation could have been better, in seven steps, we are able to determine the root of this function, 1. Example 3, once again, highlights the versatility of the Newton-Raphson method. Although this function is simple, the principle that the Newton-Raphson method can find roots of trigonometric functions shows that Newton's method extends over a wide variety of functions.

Had Function 3 been a more complex, differentiable, trigonometric function, Newton's method would have been able to determine the roots. An interesting point in this example is that the approximations (Xn+1) do not approach the root from one direction. In this example, the approximations "bounce" back and forth around the root before finding it precisely. While Newton's method helps us find the roots of these functions with ease, there are some functions to which we cannot apply it. The following examples illustrate such cases. In example 4, we are faced with a cubed-root function.

The table of values shows that the pattern of values for Xn+1 is divergent. They do not approach a single value, the root of the function. Had we not known the root of the function, Z = 0, we would have been misled by Newton's method. Newton's method indicates that no real root exists, when, in actuality, one does. In example 5, we are faced with a quadratic similar to Function 1. Newton's method fails to find the root in this case because the tangent line at the root has a slope of zero.

Thus, we are faced with the problem of dividing by zero. Newton's method is most useful in finding roots of polynomial functions of degrees higher than 4 and complex trigonometric functions. In cases of linear, quadratic, cubic, and quartic functions, it is more prudent to use algebraic formulas in finding roots. In the case of simple trigonometric functions, inverse trigonometric functions serve as a more efficient way to find roots.

Bibliography

Newton Basins. August, 1994.
Clark University. November 19th, 2001 web's Method And Fractals.
Undated. University of Hawaii. November 19th, 2001 web A.
S. The Numerical Treatment of a Single Nonlinear Equation. New York: McGraw-Hill, 1970.
Newton, I. METHODS FLUXION UM ET SERIE RUM INFINITARUM, 1664-1671. Thomas, G.B. & Finney, R.L. Calculus and Analytic Geometry, Reading, Massachusetts: Addison, Wesley, and Longman, 1982.