I think there are other better ways...
f = @(x) (tan((31.157/180) * pi) * x) - ((9.81*x.^2)/(2*25*25*(cos((31.157/180)*pi))^2));
g = @(x) (tan((62.774/180) * pi) * x) - ((9.81*x.^2)/(2*25*25*cos(((62.774/180)*pi))^2));
% tangent gradient at origin
syms x
df = matlabFunction( diff(f(x)) );
dg = matlabFunction( diff(g(x)) );
alpha = atan(df(0));
beta = atan(dg(0));
x = 0:0.1:60;
plot(x, f(x));
grid on;hold on;axis equal
plot(x, g(x));
% plot(x,df(0)*x); % tangent line
% plot(x,dg(0)*x);
% circular arc
% alpha
xIntF = 5; % X-coordinate of the intersection of circle and f(x)
yIntF = f(xIntF);
rIntF = sqrt(xIntF^2+yIntF^2); % radius for alpha
theta = linspace(0,atan(yIntF/xIntF),100);
xalpha = rIntF*cos(theta);
yalpha = rIntF*sin(theta);
plot(xalpha,yalpha,'-k');
xt = 6; yt = 1; % text location
str = sprintf('\\alpha=%2.1f°',alpha*180/pi);
text(xt,yt,str)
% beta
xIntG = 6; % X-coordinate of the intersection of circle and g(x)
yIntG = g(xIntG);
rIntG = sqrt(xIntG^2+yIntG^2); % radius for beta
theta = linspace(0,atan(yIntG/xIntG),100);
xbeta = rIntG*cos(theta);
ybeta = rIntG*sin(theta);
plot(xbeta,ybeta,'-k');
xt = 13; yt = 1; % text location
str = sprintf('\\beta=%2.1f°',beta*180/pi);
text(xt,yt,str)
