Does $cosleft(frac{2pi}2right) = 0$?
I am studying chapter 10 (Partial Differential Equations and Fourier Series) of Boyce's Elementary Differential Equations and I stumbled upon this question:
Consider the conduction of heat in a rod $40~rm cm$ in length whose ends are maintained at $0^circrm C$ for
all $t > 0$. In the below question find an expression for the temperature $u(x, t)$ if the initial temperature distribution in the rod is the given function. Suppose that $α_2 = 1$.
$u(x, 0) = begin{cases}
x & 0 le x lt20\
40-x & 20 le x le40\
end{cases}
\$
While I was checking the soluotion of this question I found online I saw in one of the steps involved below is treated as zero:
$$cosfrac{frac{(npi L)}{2}}{L} = 0.$$
I just wish to ask why is that the case that the above equals zero? For instance if $n = 2$ shouldn't the above equals $cos(pi) = -1$?
Thank you!
*this link is a screenshot of the solution I found online:
!https://imgur.com/pLHwZHe ; link to the full solution is here: !https://math.berkeley.edu/~ogus/old/Math_54-05/HW%20solutions/homework509.pdf
trigonometry pde fourier-series
edited Nov 26 at 15:19
Davide Giraudo
125k16150259
asked Nov 26 at 10:23
Lullaby
42
add a comment |
I am studying chapter 10 (Partial Differential Equations and Fourier Series) of Boyce's Elementary Differential Equations and I stumbled upon this question:
Consider the conduction of heat in a rod $40~rm cm$ in length whose ends are maintained at $0^circrm C$ for
all $t > 0$. In the below question find an expression for the temperature $u(x, t)$ if the initial temperature distribution in the rod is the given function. Suppose that $α_2 = 1$.
$u(x, 0) = begin{cases}
x & 0 le x lt20\
40-x & 20 le x le40\
end{cases}
\$
While I was checking the soluotion of this question I found online I saw in one of the steps involved below is treated as zero:
$$cosfrac{frac{(npi L)}{2}}{L} = 0.$$
I just wish to ask why is that the case that the above equals zero? For instance if $n = 2$ shouldn't the above equals $cos(pi) = -1$?
Thank you!
*this link is a screenshot of the solution I found online:
!https://imgur.com/pLHwZHe ; link to the full solution is here: !https://math.berkeley.edu/~ogus/old/Math_54-05/HW%20solutions/homework509.pdf
trigonometry pde fourier-series
edited Nov 26 at 15:19
Davide Giraudo
125k16150259
asked Nov 26 at 10:23
Lullaby
42
1
Your links say $cos frac{npi L/2}{L}$, not what you wrote.
– Ennar
Nov 26 at 10:38
Thanks for pointing it out I think it's changed by Tianlalu while he's correcting a bunch of formatting error I made... let me change it back thanks both of you
– Lullaby
Nov 26 at 10:43
Ah, I see, yes. I don't understand why they would deliberately change the meaning without confirming with you. As to the question, the expression is $0$ for odd $n$.
– Ennar
Nov 26 at 10:48
$n$ is an odd integer.
– Alex Silva
Nov 26 at 10:48
add a comment |
I am studying chapter 10 (Partial Differential Equations and Fourier Series) of Boyce's Elementary Differential Equations and I stumbled upon this question:
Consider the conduction of heat in a rod $40~rm cm$ in length whose ends are maintained at $0^circrm C$ for
all $t > 0$. In the below question find an expression for the temperature $u(x, t)$ if the initial temperature distribution in the rod is the given function. Suppose that $α_2 = 1$.
$u(x, 0) = begin{cases}
x & 0 le x lt20\
40-x & 20 le x le40\
end{cases}
\$
While I was checking the soluotion of this question I found online I saw in one of the steps involved below is treated as zero:
$$cosfrac{frac{(npi L)}{2}}{L} = 0.$$
I just wish to ask why is that the case that the above equals zero? For instance if $n = 2$ shouldn't the above equals $cos(pi) = -1$?
Thank you!
*this link is a screenshot of the solution I found online:
!https://imgur.com/pLHwZHe ; link to the full solution is here: !https://math.berkeley.edu/~ogus/old/Math_54-05/HW%20solutions/homework509.pdf
trigonometry pde fourier-series
edited Nov 26 at 15:19
Davide Giraudo
125k16150259
asked Nov 26 at 10:23
Lullaby
42
I am studying chapter 10 (Partial Differential Equations and Fourier Series) of Boyce's Elementary Differential Equations and I stumbled upon this question:
Consider the conduction of heat in a rod $40~rm cm$ in length whose ends are maintained at $0^circrm C$ for
all $t > 0$. In the below question find an expression for the temperature $u(x, t)$ if the initial temperature distribution in the rod is the given function. Suppose that $α_2 = 1$.
$u(x, 0) = begin{cases}
x & 0 le x lt20\
40-x & 20 le x le40\
end{cases}
\$
While I was checking the soluotion of this question I found online I saw in one of the steps involved below is treated as zero:
$$cosfrac{frac{(npi L)}{2}}{L} = 0.$$
I just wish to ask why is that the case that the above equals zero? For instance if $n = 2$ shouldn't the above equals $cos(pi) = -1$?
Thank you!
*this link is a screenshot of the solution I found online:
!https://imgur.com/pLHwZHe ; link to the full solution is here: !https://math.berkeley.edu/~ogus/old/Math_54-05/HW%20solutions/homework509.pdf
trigonometry pde fourier-series
trigonometry pde fourier-series
edited Nov 26 at 15:19
Davide Giraudo
125k16150259
asked Nov 26 at 10:23
Lullaby
42
edited Nov 26 at 15:19
Davide Giraudo
125k16150259
asked Nov 26 at 10:23
Lullaby
42
edited Nov 26 at 15:19
Davide Giraudo
125k16150259
edited Nov 26 at 15:19
Davide Giraudo
125k16150259
edited Nov 26 at 15:19
Davide Giraudo
125k16150259
125k16150259
asked Nov 26 at 10:23
Lullaby
42
asked Nov 26 at 10:23
Lullaby
42
asked Nov 26 at 10:23
Lullaby
42
42
1
Your links say $cos frac{npi L/2}{L}$, not what you wrote.
– Ennar
Nov 26 at 10:38
Thanks for pointing it out I think it's changed by Tianlalu while he's correcting a bunch of formatting error I made... let me change it back thanks both of you
– Lullaby
Nov 26 at 10:43
Ah, I see, yes. I don't understand why they would deliberately change the meaning without confirming with you. As to the question, the expression is $0$ for odd $n$.
– Ennar
Nov 26 at 10:48
$n$ is an odd integer.
– Alex Silva
Nov 26 at 10:48
add a comment |
1
Your links say $cos frac{npi L/2}{L}$, not what you wrote.
– Ennar
Nov 26 at 10:38
Thanks for pointing it out I think it's changed by Tianlalu while he's correcting a bunch of formatting error I made... let me change it back thanks both of you
– Lullaby
Nov 26 at 10:43
Ah, I see, yes. I don't understand why they would deliberately change the meaning without confirming with you. As to the question, the expression is $0$ for odd $n$.
– Ennar
Nov 26 at 10:48
$n$ is an odd integer.
– Alex Silva
Nov 26 at 10:48
1
1
Your links say $cos frac{npi L/2}{L}$, not what you wrote.
– Ennar
Nov 26 at 10:38
Your links say $cos frac{npi L/2}{L}$, not what you wrote.
– Ennar
Nov 26 at 10:38
Thanks for pointing it out I think it's changed by Tianlalu while he's correcting a bunch of formatting error I made... let me change it back thanks both of you
– Lullaby
Nov 26 at 10:43
Thanks for pointing it out I think it's changed by Tianlalu while he's correcting a bunch of formatting error I made... let me change it back thanks both of you
– Lullaby
Nov 26 at 10:43
Ah, I see, yes. I don't understand why they would deliberately change the meaning without confirming with you. As to the question, the expression is $0$ for odd $n$.
– Ennar
Nov 26 at 10:48
Ah, I see, yes. I don't understand why they would deliberately change the meaning without confirming with you. As to the question, the expression is $0$ for odd $n$.
– Ennar
Nov 26 at 10:48
$n$ is an odd integer.
– Alex Silva
Nov 26 at 10:48
$n$ is an odd integer.
– Alex Silva
Nov 26 at 10:48
add a comment |
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1
Your links say $cos frac{npi L/2}{L}$, not what you wrote.
– Ennar
Nov 26 at 10:38
Thanks for pointing it out I think it's changed by Tianlalu while he's correcting a bunch of formatting error I made... let me change it back thanks both of you
– Lullaby
Nov 26 at 10:43
Ah, I see, yes. I don't understand why they would deliberately change the meaning without confirming with you. As to the question, the expression is $0$ for odd $n$.
– Ennar
Nov 26 at 10:48
$n$ is an odd integer.
– Alex Silva
Nov 26 at 10:48