Pre-Calc Exam Notes 133

Pre-Calc Exam Notes 133 - There is a large ±eld of...

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Numerical Methods in Trigonometry Section 6.2 133 6.2 Numerical Methods in Trigonometry We were able to solve the trigonometric equations in the previous section fairly easily, which in general is not the case. For example, consider the equation cos x = x . (6.1) There is a solution, as shown in Figure 6.2.1 below. The graphs of y = cos x and y = x intersect somewhere between x = 0 and x = 1, which means that there is an x in the interval [0 , 1] such that cos x = x . -4 -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 y x cos( x ) x Figure 6.2.1 y = cos x and y = x Unfortunately there is no trigonometric identity or simple method which will help us here. Instead, we have to resort to numerical methods , which provide ways of getting successively better approximations to the actual solution(s) to within any desired degree of accuracy.
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Unformatted text preview: There is a large ±eld of mathematics devoted to this subject called numerical analysis . Many of the methods require calculus, but luckily there is a method which we can use that requires just basic algebra. It is called the secant method , and it ±nds roots of a given function f ( x ), i.e. values of x such that f ( x ) = 0. A derivation of the secant method is beyond the scope of this book, 1 but we can state the algorithm it uses to solve f ( x ) = 0: 1 For an explanation of why the secant method works, see pp. 338-344 in A. RALSTON AND P. RABINOWITZ, A First Course in Numerical Analysis , 2nd ed., New York: McGraw-Hill Book Co., 1978....
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