Calculate the first four sexagesimal places of the approximation to x=Sin 1^∘ following the meth

Calculate the first four sexagesimal places of the...

Key Concepts

Trigonometric Function Approximations

Trigonometric function approximations, such as those using Taylor series or other methods, are essential for calculating functions like sin 1°. These approximations allow one to replace the trigonometric function with a polynomial that approximates the function to a desired degree of accuracy. Understanding this concept is key to applying iterative methods and error analyses to compute values with multiple significant digits in different numeral systems.

Sexagesimal Notation

Sexagesimal notation is a numerical system based on the number 60, historically used for angles and time. In the context of measuring angles, it divides a degree into 60 minutes and each minute into 60 seconds. Understanding this system is essential when the problem requires expressing a number as a combination of degrees, arc minutes, and arc seconds, ensuring the approximation is communicated in the traditional angular format.

Iterative Approximation Methods

Iterative approximation methods involve using a repeated process to converge on a solution. By starting with an initial guess, a sequence of calculations refines the estimate, drawing it closer to the true value with each iteration. This method is particularly useful when direct computation is complex or when approximations are sufficient for practical purposes, as it can allow one to compute values like sin 1° with a quantifiable error reduction at each step.

Convergence and the Contraction Mapping Principle

The convergence of an iterative method often depends on the contraction mapping principle, which assures that if a function reduces the distance between successive points (i.e., it is a contraction), then the iteration will converge to a unique fixed point. This mathematical guarantee is crucial when applying an iterative algorithm, as it provides the theoretical foundation for why the successive approximations approach the actual value reliably.

Transcript

00:01 We will use both newton's method and the second method to calculate root for the nonlinear equation sign of x equals 0.

00:10 As first guess for the newton's method, we got to use x0 equals 1.

00:16 And as the other guess to the second method, beside this x not equal 1, is we got to use the first guess from newton's method.

00:29 So we get to apply first the newton's method.

00:32 To obtain its first guess and use that value besides the value one as the first guesses for the second method.

00:47 Now here we know that the newton's method is given by xub n plus 1 equals x of n minus f of x of n divided by the derivative of f at x of n for n greater than equal to 0 and x x not given.

01:14 And for the second method, we have x sub n equals x of n minus 1 minus f of x of n minus 1 times a fraction whose numerator is x of n minus 1 minus 1 minus x of n minus 2 divided by f sub x sub n minus 1 minus f at x of n minus 2 and that for n very then are equal to 2 because we can we begin for you start calculating x2 from x0 and x1 which are given and then for the second method we only need the function f and for the newton's method we need the function f and its derivative.

02:23 And the function f, we know in this case, is equal to sine of x.

02:29 His derivative is cosine of x.

02:33 So these are the two functions we use in the newton's method.

02:37 And this is the function we got to use in the second method to write a computer program or an updating rule in a calculator...

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