MHB Boost Your Cosine-Sine Isotherm Sketching Skills | Expert Tips & Guidance

[brunette15]

I am trying to sketch isotherms of the field cos(x)sinh(y). I am not sure how to begin with this. Can someone please help/hint me through what i have to do?

 

[Ackbach]

Presumably, you actually have a temperature field, right? That is, $T=\cos(x)\sinh(y)$. In that case, you want to sketch whole bunch of curves in the $(x,y)$ plane of the form $C=\cos(x)\sinh(y)$. Why a constant? Because you're after the isotherms - the prefix "iso" meaning "same". In fact, you can solve this equation for $y$:
\begin{align*}
C&=\cos(x)\sinh(y) \\
\sinh(y)&=C \sec(x) \\
y&=\text{arcsinh}(C\sec(x)).
\end{align*}
So, pick a few $C$'s, plot the above function, and you've got your isotherms.

 

[brunette15]

Presumably, you actually have a temperature field, right? That is, $T=\cos(x)\sinh(y)$. In that case, you want to sketch whole bunch of curves in the $(x,y)$ plane of the form $C=\cos(x)\sinh(y)$. Why a constant? Because you're after the isotherms - the prefix "iso" meaning "same". In fact, you can solve this equation for $y$:
\begin{align*}
C&=\cos(x)\sinh(y) \\
\sinh(y)&=C \sec(x) \\
y&=\text{arcsinh}(C\sec(x)).
\end{align*}
So, pick a few $C$'s, plot the above function, and you've got your isotherms.

Thankyou so much :)