From the graph I observed that these 3 functions cut at a particular point.
but what should be the value of x which will satisfy the given eqn I am unable to figure it out.
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Welcome Aniruth. $y=arc \cos x=\cos x$ so $\cos y=x$ which symmetric with $\cos x =y$, that is $x=y$, no need for third equation i,e, $y=x$ – sirous Aug 24 '18 at 16:14
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@Anirudh Please recall that if the OP is solved you can evaluate to accept an answer among the given, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work – user Sep 17 '18 at 20:11
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Since $\cos^{-1}x$ is the inverse of $\cos x$ their intersection must be at the point such that
$$\cos x=x$$
which is a trascendental equation discussed here What is the solution of cos(x)=x?.
user
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You want to solve $f(x)=0$ with $f(x):=x-\cos x$. This can only be done numerically. Since $g(x):=x-\frac{f}{f'}=\frac{x\sin x+\cos x}{1+\sin x}$, we can approximate the root with the Newton-Raphson method. The root looks close to $1$ to me, so define $x_0:=1,\,x_{n+1}:=g(x_n)$. The sequence $x_n$ continues 0.75036386784, 0.739112890911, 0.739085133385, 0.739085133215, 0.739085133215. That gets us the result to as many decimal places as Python gave me.
J.G.
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