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Solve an equation [closed]
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I want to solve the following equation for $t:$
$ z = a_1 tan(t) + a_2 tan(a_3 t)$,
where $a_1, a_2, a_3$ are constant.
I tried writing $tan$ as $sin/cos,$ frankly that's the only "idea" I had so far.
analysis systems-of-equations
edited Jul 20 at 22:08
Adrian Keister
3,61721533
asked Jul 20 at 22:02
Diamir
88111
closed as off-topic by amWhy, Xander Henderson, John Ma, Shailesh, Parcly Taxel Jul 21 at 10:11
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Xander Henderson, John Ma, Shailesh, Parcly Taxel
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up vote
1
down vote
favorite
I want to solve the following equation for $t:$
$ z = a_1 tan(t) + a_2 tan(a_3 t)$,
where $a_1, a_2, a_3$ are constant.
I tried writing $tan$ as $sin/cos,$ frankly that's the only "idea" I had so far.
analysis systems-of-equations
edited Jul 20 at 22:08
Adrian Keister
3,61721533
asked Jul 20 at 22:02
Diamir
88111
closed as off-topic by amWhy, Xander Henderson, John Ma, Shailesh, Parcly Taxel Jul 21 at 10:11
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Xander Henderson, John Ma, Shailesh, Parcly Taxel
Most equations in mathematics aren't solvable, in the sense of being able to get a formula for the solution. You'd do better to ask why you want to solve it - what information do you need about the solution?
– Jack M
Jul 21 at 8:47
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I want to solve the following equation for $t:$
$ z = a_1 tan(t) + a_2 tan(a_3 t)$,
where $a_1, a_2, a_3$ are constant.
I tried writing $tan$ as $sin/cos,$ frankly that's the only "idea" I had so far.
analysis systems-of-equations
edited Jul 20 at 22:08
Adrian Keister
3,61721533
asked Jul 20 at 22:02
Diamir
88111
I want to solve the following equation for $t:$
$ z = a_1 tan(t) + a_2 tan(a_3 t)$,
where $a_1, a_2, a_3$ are constant.
I tried writing $tan$ as $sin/cos,$ frankly that's the only "idea" I had so far.
analysis systems-of-equations
edited Jul 20 at 22:08
Adrian Keister
3,61721533
asked Jul 20 at 22:02
Diamir
88111
edited Jul 20 at 22:08
Adrian Keister
3,61721533
edited Jul 20 at 22:08
Adrian Keister
3,61721533
edited Jul 20 at 22:08
Adrian Keister
3,61721533
3,61721533
asked Jul 20 at 22:02
Diamir
88111
asked Jul 20 at 22:02
Diamir
88111
asked Jul 20 at 22:02
Diamir
88111
88111
closed as off-topic by amWhy, Xander Henderson, John Ma, Shailesh, Parcly Taxel Jul 21 at 10:11
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Xander Henderson, John Ma, Shailesh, Parcly Taxel
closed as off-topic by amWhy, Xander Henderson, John Ma, Shailesh, Parcly Taxel Jul 21 at 10:11
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Xander Henderson, John Ma, Shailesh, Parcly Taxel
Most equations in mathematics aren't solvable, in the sense of being able to get a formula for the solution. You'd do better to ask why you want to solve it - what information do you need about the solution?
– Jack M
Jul 21 at 8:47
add a comment |Â
Most equations in mathematics aren't solvable, in the sense of being able to get a formula for the solution. You'd do better to ask why you want to solve it - what information do you need about the solution?
– Jack M
Jul 21 at 8:47
Most equations in mathematics aren't solvable, in the sense of being able to get a formula for the solution. You'd do better to ask why you want to solve it - what information do you need about the solution?
– Jack M
Jul 21 at 8:47
Most equations in mathematics aren't solvable, in the sense of being able to get a formula for the solution. You'd do better to ask why you want to solve it - what information do you need about the solution?
– Jack M
Jul 21 at 8:47
add a comment |Â
1 Answer
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You must take care about the fact that there is an infinite number of zero's for the function
$$f(t) = a_1 tan(t) + a_2 tan(a_3 t)-z$$ which shows asymptotes at $(2k-1)frac pi 2$ and $(2k-1)frac pi 2a_3$ and there will be a root between two asymptotes.
When you will know which of the root you are looking for (that is to say between which pair of asymptotes - this gives upper and lower bounds for the root), you will need anumerical method such as Newton. The problem is that Newton iterations can take you away from the given bounds; so, either you use bisection or you use, for faster convergence, a combination of bisection and Newton steps.
For the later, have a look at subroutine rtsafe in "Numerical Recipes".
answered Jul 21 at 3:57
Claude Leibovici
111k1055126
Thanks. Is there also a way to solve it analytically? At least for suitable t?
– Diamir
Jul 21 at 8:18
@Diamir. If $a_3$ is an integer, it would reduce to a polynomial in $tan(t)$; that' all.
– Claude Leibovici
Jul 21 at 8:21
This is a monotonic function. Bisection will work just fine and converge very quickly, and is easier to understand/program than Newton's method.
– Jack M
Jul 21 at 9:25
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
You must take care about the fact that there is an infinite number of zero's for the function
$$f(t) = a_1 tan(t) + a_2 tan(a_3 t)-z$$ which shows asymptotes at $(2k-1)frac pi 2$ and $(2k-1)frac pi 2a_3$ and there will be a root between two asymptotes.
When you will know which of the root you are looking for (that is to say between which pair of asymptotes - this gives upper and lower bounds for the root), you will need anumerical method such as Newton. The problem is that Newton iterations can take you away from the given bounds; so, either you use bisection or you use, for faster convergence, a combination of bisection and Newton steps.
For the later, have a look at subroutine rtsafe in "Numerical Recipes".
answered Jul 21 at 3:57
Claude Leibovici
111k1055126
Thanks. Is there also a way to solve it analytically? At least for suitable t?
– Diamir
Jul 21 at 8:18
@Diamir. If $a_3$ is an integer, it would reduce to a polynomial in $tan(t)$; that' all.
– Claude Leibovici
Jul 21 at 8:21
This is a monotonic function. Bisection will work just fine and converge very quickly, and is easier to understand/program than Newton's method.
– Jack M
Jul 21 at 9:25
add a comment |Â
up vote
0
down vote
You must take care about the fact that there is an infinite number of zero's for the function
$$f(t) = a_1 tan(t) + a_2 tan(a_3 t)-z$$ which shows asymptotes at $(2k-1)frac pi 2$ and $(2k-1)frac pi 2a_3$ and there will be a root between two asymptotes.
When you will know which of the root you are looking for (that is to say between which pair of asymptotes - this gives upper and lower bounds for the root), you will need anumerical method such as Newton. The problem is that Newton iterations can take you away from the given bounds; so, either you use bisection or you use, for faster convergence, a combination of bisection and Newton steps.
For the later, have a look at subroutine rtsafe in "Numerical Recipes".
answered Jul 21 at 3:57
Claude Leibovici
111k1055126
Thanks. Is there also a way to solve it analytically? At least for suitable t?
– Diamir
Jul 21 at 8:18
@Diamir. If $a_3$ is an integer, it would reduce to a polynomial in $tan(t)$; that' all.
– Claude Leibovici
Jul 21 at 8:21
This is a monotonic function. Bisection will work just fine and converge very quickly, and is easier to understand/program than Newton's method.
– Jack M
Jul 21 at 9:25
add a comment |Â
up vote
0
down vote
up vote
0
down vote
You must take care about the fact that there is an infinite number of zero's for the function
$$f(t) = a_1 tan(t) + a_2 tan(a_3 t)-z$$ which shows asymptotes at $(2k-1)frac pi 2$ and $(2k-1)frac pi 2a_3$ and there will be a root between two asymptotes.
When you will know which of the root you are looking for (that is to say between which pair of asymptotes - this gives upper and lower bounds for the root), you will need anumerical method such as Newton. The problem is that Newton iterations can take you away from the given bounds; so, either you use bisection or you use, for faster convergence, a combination of bisection and Newton steps.
For the later, have a look at subroutine rtsafe in "Numerical Recipes".
answered Jul 21 at 3:57
Claude Leibovici
111k1055126
You must take care about the fact that there is an infinite number of zero's for the function
$$f(t) = a_1 tan(t) + a_2 tan(a_3 t)-z$$ which shows asymptotes at $(2k-1)frac pi 2$ and $(2k-1)frac pi 2a_3$ and there will be a root between two asymptotes.
When you will know which of the root you are looking for (that is to say between which pair of asymptotes - this gives upper and lower bounds for the root), you will need anumerical method such as Newton. The problem is that Newton iterations can take you away from the given bounds; so, either you use bisection or you use, for faster convergence, a combination of bisection and Newton steps.
For the later, have a look at subroutine rtsafe in "Numerical Recipes".
answered Jul 21 at 3:57
Claude Leibovici
111k1055126
edited Jul 21 at 8:20
answered Jul 21 at 3:57
Claude Leibovici
111k1055126
answered Jul 21 at 3:57
Claude Leibovici
111k1055126
answered Jul 21 at 3:57
Claude Leibovici
111k1055126
111k1055126
Thanks. Is there also a way to solve it analytically? At least for suitable t?
– Diamir
Jul 21 at 8:18
@Diamir. If $a_3$ is an integer, it would reduce to a polynomial in $tan(t)$; that' all.
– Claude Leibovici
Jul 21 at 8:21
This is a monotonic function. Bisection will work just fine and converge very quickly, and is easier to understand/program than Newton's method.
– Jack M
Jul 21 at 9:25
add a comment |Â
Thanks. Is there also a way to solve it analytically? At least for suitable t?
– Diamir
Jul 21 at 8:18
@Diamir. If $a_3$ is an integer, it would reduce to a polynomial in $tan(t)$; that' all.
– Claude Leibovici
Jul 21 at 8:21
This is a monotonic function. Bisection will work just fine and converge very quickly, and is easier to understand/program than Newton's method.
– Jack M
Jul 21 at 9:25
Thanks. Is there also a way to solve it analytically? At least for suitable t?
– Diamir
Jul 21 at 8:18
Thanks. Is there also a way to solve it analytically? At least for suitable t?
– Diamir
Jul 21 at 8:18
@Diamir. If $a_3$ is an integer, it would reduce to a polynomial in $tan(t)$; that' all.
– Claude Leibovici
Jul 21 at 8:21
@Diamir. If $a_3$ is an integer, it would reduce to a polynomial in $tan(t)$; that' all.
– Claude Leibovici
Jul 21 at 8:21
This is a monotonic function. Bisection will work just fine and converge very quickly, and is easier to understand/program than Newton's method.
– Jack M
Jul 21 at 9:25
This is a monotonic function. Bisection will work just fine and converge very quickly, and is easier to understand/program than Newton's method.
– Jack M
Jul 21 at 9:25
add a comment |Â
Most equations in mathematics aren't solvable, in the sense of being able to get a formula for the solution. You'd do better to ask why you want to solve it - what information do you need about the solution?
– Jack M
Jul 21 at 8:47