Question

Using format long g, verify the following trigonometric identities by evaluating the left- and right-hand sides independently, c) arcsin x + arcsin y = arcsin(x/1 - y2 + yv1 – x2) for x = 0.5, y = 13/2 d) arctan x + arctan y = arctan (1971) for x = 0.5, y = 13/2 Hint: You must assign values to variables before using them in a command. Then, for each identity, assign the left hand side to a variable name LHS and the right hand side to a variable name RHS and calculate the value of each variable separately, then calculate the absolute difference between them and verify it is almost zero, i.e., İLHS - RHS| < 10-12 Use format long for this problem. Your answer should show that the value calculated for each variable is exactly the same to at least 12 digits. You can use the underscores _a for part a) and _b for part b), and so on. You must notice that the + in parts c) and d) means that you must verify each identity twice, one for the + sign and one for the - sign. You must notice the difference between using the angle in radians and in degree and the proper trigonometric built-in function in MATLAB to use in each case. Display only the values of variables that are required to be compared to each other.

Answer 1

% Matlab script to verify trignometric identities by evaluating the
% left-hand and right-hand sides independently

format long g;
format compact;
% define the x and y values
x = 0.5;
y = sqrt(3)/2;
% a(i)
% compute LHS and RHS independently
LHS = asin(x) + asin(y);
RHS = asin((x*sqrt(1-y^2)) + (y*sqrt(1-x^2)));
% calculate the absolute difference between LHS and RHS
diff = abs(LHS - RHS);
% display the LHS, RHS and absolute difference
disp("LHS: ");
disp(LHS);
disp("RHS: ");
disp(RHS);
disp("Absolute difference: ");
disp(diff);

% a(ii)
LHS = asin(x) - asin(y);
RHS = asin((x*sqrt(1-y^2)) - (y*sqrt(1-x^2)));

diff = abs(LHS - RHS);
disp("LHS: ");
disp(LHS);
disp("RHS: ");
disp(RHS);
disp("Absolute difference: ");
disp(diff);

% b(i)
LHS = atan(x) + atan(y);
RHS = atan((x+y)/(1-(x*y)));
diff = abs(LHS-RHS);
disp("LHS: ");
disp(LHS);
disp("RHS: ");
disp(RHS);
disp("Absolute difference: ");
disp(diff);

% b(ii)
LHS = atan(x) - atan(y);
RHS = atan((x-y)/(1+(x*y)));
diff = abs(LHS-RHS);
disp("LHS: ");
disp(LHS);
disp("RHS: ");
disp(RHS);
disp("Absolute difference: ");
disp(diff);

%end of script

Output:

LHS: 1.5707963267949 RHS: 1.5707963267949 Absolute difference: 0 LHS: -0.523598775598299 RHS: -0.523598775598299 Absolute difference: LHS: 1.17737198794557 RHS: 1.17737198794557 Absolute difference: LHS: -0.250076769943959 RHS: -0.250076769943959 Absolute difference: