For what value of $x$ $x$ does $sin (x) + cos (x) = x$ $sin(x)+cos(x)=x$ ?

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For what value of $x$ $x$ does $sin (x) + cos (x) = x$ $sin(x)+cos(x)=x$ ?

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Answer 1 · 2017-11-06T12:40:19

Assuming we are working in radians then
$XXX x \approx 1.2587$ $color(white)("XXX")x~~1.2587$

Explanation:

I suspect this question is invalid, but in case it is not (and in order to clear it from the unanswered question list), here is what I was able to come up with:

If I graph $f (x) = sin (x) + cos (x)$ $f(x)=sin(x)+cos(x)$ and $f (x) = x$ $f(x)=x$
I get an intersection point that is between $x = 1.2$ $x=1.2$ and $x = 1.3$ $x=1.3$
enter image source here
Using a spreadsheet and a variant of the Newton Method I was able to approximate this more closely at about $x = 1.2587$ $x=1.2587$

NOTE This is not intended to be a "proper" solution. Perhaps there is some way to use the Taylor Expansion for $arctan$ $arctan$ to get a better solution but, since I was uncertain that the question was even correct, this is the best I could do without excessive analysis.

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For what value of $x$ $x$ does $sin (x) + cos (x) = x$ $sin(x)+cos(x)=x$ ?

1 Answer

Explanation:

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For what value of xxx does sin(x)+cos(x)=xsin(x)+cos(x)=xsin(x)+cos(x)=x?

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For what value of $x$ $x$ does $sin (x) + cos (x) = x$ $sin(x)+cos(x)=x$ ?