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Bound on sine of sum of angles
Maximum Value of Trig ExpressionHow to find maximum and minimum volumes of solid obtained by rotating $y=sin x$ around $y=c$Find the maximum and minimum values of $A cos t + B sin t$Maxima and Minima of Sin(x)/xConvolution of sine and cosine.Extrema of ellipse from parametric formSecond derivative test $sin x + sin y + sin(x-y)$Calculus inequality involving sine and cosineVolume of the solid bounded by the region $E = (x,y,z) , : , x^2 + y^2 +z^2 - 2z leq 0, sqrtx^2+y^2 leq z$Questions on changing bounds of integration for double-integrals
$begingroup$
I would like to find tight bounds of the form
$$
f_l(x) g_l(y) leq sin( x + y) leq f_u(x) g_u(y)
$$
Especially for $x in [0, pi / N]$ and $y in [k pi / N]$ where $k$ and $N$ are integers and $|k| < N/2$. I realize that for a given $N$ the choice of $y$ is finite and therefore tight bounds can be established by taking the maximum or minimum over the set of all possibilities. However, if possible I would appreciate a more analytical approach / formulation.
For example, the identity $sin( x+ y) = sin(x) cos(y) + cos(x) sin(y)$ can be used to set some bounds by setting $x$ to the minimum and maximum values.
real-analysis calculus
$endgroup$
add a comment |
$begingroup$
I would like to find tight bounds of the form
$$
f_l(x) g_l(y) leq sin( x + y) leq f_u(x) g_u(y)
$$
Especially for $x in [0, pi / N]$ and $y in [k pi / N]$ where $k$ and $N$ are integers and $|k| < N/2$. I realize that for a given $N$ the choice of $y$ is finite and therefore tight bounds can be established by taking the maximum or minimum over the set of all possibilities. However, if possible I would appreciate a more analytical approach / formulation.
For example, the identity $sin( x+ y) = sin(x) cos(y) + cos(x) sin(y)$ can be used to set some bounds by setting $x$ to the minimum and maximum values.
real-analysis calculus
$endgroup$
$begingroup$
What kind of bounds? $f_lequiv -1, g_lequiv 1$ and $f_u=g_uequiv 1$ works.
$endgroup$
– mfl
Mar 11 at 20:04
$begingroup$
@mfl as tight as possible bounds for the given values.
$endgroup$
– Sebastian Schlecht
Mar 11 at 20:06
add a comment |
$begingroup$
I would like to find tight bounds of the form
$$
f_l(x) g_l(y) leq sin( x + y) leq f_u(x) g_u(y)
$$
Especially for $x in [0, pi / N]$ and $y in [k pi / N]$ where $k$ and $N$ are integers and $|k| < N/2$. I realize that for a given $N$ the choice of $y$ is finite and therefore tight bounds can be established by taking the maximum or minimum over the set of all possibilities. However, if possible I would appreciate a more analytical approach / formulation.
For example, the identity $sin( x+ y) = sin(x) cos(y) + cos(x) sin(y)$ can be used to set some bounds by setting $x$ to the minimum and maximum values.
real-analysis calculus
$endgroup$
I would like to find tight bounds of the form
$$
f_l(x) g_l(y) leq sin( x + y) leq f_u(x) g_u(y)
$$
Especially for $x in [0, pi / N]$ and $y in [k pi / N]$ where $k$ and $N$ are integers and $|k| < N/2$. I realize that for a given $N$ the choice of $y$ is finite and therefore tight bounds can be established by taking the maximum or minimum over the set of all possibilities. However, if possible I would appreciate a more analytical approach / formulation.
For example, the identity $sin( x+ y) = sin(x) cos(y) + cos(x) sin(y)$ can be used to set some bounds by setting $x$ to the minimum and maximum values.
real-analysis calculus
real-analysis calculus
edited Mar 11 at 20:18
Sebastian Schlecht
asked Mar 11 at 19:59
Sebastian SchlechtSebastian Schlecht
23718
23718
$begingroup$
What kind of bounds? $f_lequiv -1, g_lequiv 1$ and $f_u=g_uequiv 1$ works.
$endgroup$
– mfl
Mar 11 at 20:04
$begingroup$
@mfl as tight as possible bounds for the given values.
$endgroup$
– Sebastian Schlecht
Mar 11 at 20:06
add a comment |
$begingroup$
What kind of bounds? $f_lequiv -1, g_lequiv 1$ and $f_u=g_uequiv 1$ works.
$endgroup$
– mfl
Mar 11 at 20:04
$begingroup$
@mfl as tight as possible bounds for the given values.
$endgroup$
– Sebastian Schlecht
Mar 11 at 20:06
$begingroup$
What kind of bounds? $f_lequiv -1, g_lequiv 1$ and $f_u=g_uequiv 1$ works.
$endgroup$
– mfl
Mar 11 at 20:04
$begingroup$
What kind of bounds? $f_lequiv -1, g_lequiv 1$ and $f_u=g_uequiv 1$ works.
$endgroup$
– mfl
Mar 11 at 20:04
$begingroup$
@mfl as tight as possible bounds for the given values.
$endgroup$
– Sebastian Schlecht
Mar 11 at 20:06
$begingroup$
@mfl as tight as possible bounds for the given values.
$endgroup$
– Sebastian Schlecht
Mar 11 at 20:06
add a comment |
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$begingroup$
What kind of bounds? $f_lequiv -1, g_lequiv 1$ and $f_u=g_uequiv 1$ works.
$endgroup$
– mfl
Mar 11 at 20:04
$begingroup$
@mfl as tight as possible bounds for the given values.
$endgroup$
– Sebastian Schlecht
Mar 11 at 20:06