%% Roots Using Substitution
% You can solve polynomial equations involving trigonometric functions by simplifying the equation using a substitution. The resulting polynomial of one variable no longer contains any trigonometric functions.
% For example, find the values of \theta that solve the equation
%  3 \cos^2(\theta) - \sin(\theta) + 3 = 0.
% Use the fact that \cos^2(\theta) = 1 - \sin^2(\theta) to express the equation entirely in terms of sine functions:
% -3\sin^2(\theta) - \sin(\theta) + 6 = 0.
% Use the substitution x = \sin(\theta) to express the equation as a simple polynomial equation:
%  -3x^2 - x + 6 = 0.
% Create a vector to represent the polynomial.

p = [-3 -1 6];

%
% Find the roots of the polynomial.

r = roots(p)

%
% To undo the substitution, use \theta = \sin^{-1}(x). The asin function calculates the inverse sine.

theta = asin(r)

%
% Verify that the elements in theta are the values of \theta that solve the original equation (within roundoff error).

f = @(Z) 3*cos(Z).^2 - sin(Z) + 3;
f(theta)

% Copyright 2015 The MathWorks, Inc.