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1. [45 pts] Part 1 - "For" and "While" loops in MATLAB Consider the approximation for sin(x) given in homework #2 as: 3 9 13 15 17 sin x = x 3! 5!7!9! 11 13 15!17! Create a vector where each element is a term of the series. Allow the user to input the value of x (as a scalar). Sum the terms of the series using a "for" loop and create a formatted output to show the user (1) the value of the approximation, (2) the actual value of sin(x) (from the MATLAB function) and (3) the error between the two. Use values x 2, 2.5, 3 to compare. ii.Now consider the generalized form of the Taylor series for sin(x): 2n+1 sinx = (-1)" n=0 Sum the first 100 terms of the series and show the result as a formatted output identical to that of 1(i). Allow the user to input the value of x as a scalar. Repeat this problem with the first 250 terms. Compare the results. (Use same x values as from (i))

Explanation / Answer

Answer:

(i) # MatLab program for given sin(x) funtion:

clc;

close all;

clear all;

format long;

x = input('Enter x value: ');

sinx = 0;

k = 2;

for i=1:2:17

sinx = sinx + (-1).^k*(x.^i)/factorial(i);

k = k + 1;

end

fprintf('The value of approximation is: %d ',sinx);

fprintf('The original value of sin(%.2f) is:%d ',x,sin(x));

fprintf('The error is: %d ',sinx-sin(x));

Output:

Enter x value: 2

The value of approximation is: 9.092974e-01

The original value of sin(2.00) is:9.092974e-01

The error is: 4.269252e-12

Enter x value: 2.5

The value of approximation is: 5.984721e-01

The original value of sin(2.50) is:5.984721e-01

The error is: 2.946692e-10

Enter x value: 3

The value of approximation is: 1.411200e-01

The original value of sin(3.00) is:1.411200e-01

The error is: 9.353375e-09

(ii) % matlab code

sumSinx = 0;

x = input("Enter x: ");

n = input("Enter n: ");

for i=0:n

sumSinx = sumSinx + ((-1)^i)*(x^(2*i+1)/(factorial(2*i +1)));

end

fprintf("n: %d x: %f sinx = %0.10f ",n,x,sumSinx);

% end matlab code

%{

Output:

Enter x: 2

Enter n: 100

n: 100 x: 2.000000 sinx = 0.9092974268

Enter x: 2

Enter n: 250

n: 250 x: 2.000000 sinx = 0.9092974268

Enter x: 2.5

Enter n: 250

n: 250 x: 2.500000 sinx = 0.5984721441

Enter x: 3

Enter n: 250

n: 250 x: 3.000000 sinx = 0.1411200081

%}

(iii) # MatLab program

clc;

close all;

clear all;

format long;

x=60;

%conversion of degree to radians

x=x*pi/180;

S1=sin(x);

num=0;

S2=0;

error=100;

%execute Loop until the 1E-4

while error>1E-4

S2=S2+((-1)^num*x^(2*num+1)/factorial(2*num+1));

error=abs((S1-S2)/S1)*100;

num=num+1;

I(num)=num;

E(num)=error;

end

%Display the result

fprintf(' Iteration:%f Number of terms in the series:%f Error is:%f ',I,I,E);

%Plot the graph iteration vs. error

plot(I,E);

Output:

$octave -qf --no-window-system main.m

Iteration: 1.000000

Number of terms in the series: 2.000000

Error is: 3.000000

Iteration: 4.000000

Number of terms in the series: 5.000000

Error is: 1.000000

Iteration: 2.000000

Number of terms in the series: 3.000000

Error is: 4.000000

Iteration: 5.000000

Number of terms in the series: 20.919958

Error is: 1.180638

Iteration: 0.031163

Number of terms in the series: 0.000477

Error is: 0.000005

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