Solution to a system of transcendental equations

[Soumalya]

I got stuck while solving a problem in statics unable to solve a system of equations of the form obtained as below,

a cosθ - b sinθ = c₁
a sinθ + b cosθ = c₂

I wish to obtain a and b in terms of c₁ and c₂.Squaring and adding I obtained,

a² + b² = c₁² + c₂²

I need another relation in a,b,c₁ and c₂ to achieve the result.How do I proceed next?

 

[PeroK]

a and b will depend not only on c_1 and c_2 but on θ.

Think about a rotation of θ about the origin.

 

[Soumalya]

a and b will depend not only on c_1 and c_2 but on θ.

Think about a rotation of θ about the origin.

Well I did forget to mention c₁ and c₂ are functions in θ.How do I solve for a and b in terms of c₁, c₂ and θ?

 

[PeroK]

Well I did forget to mention c₁ and c₂ are functions in θ.How do I solve for a and b in terms of c₁, c₂ and θ?

If you know a little linear algebra, you can express the system of equations in matrix form, then multiple by the inverse rotation matrix.

 

[Soumalya]

If you know a little linear algebra, you can express the system of equations in matrix form, then multiple by the inverse rotation matrix.

Well I am not quite used to linear algebra in practice but I will have a look if things appear easier in comparison to conventional algebra.

 

[PeroK]

3

Well I am not quite used to linear algebra in practice but I will have a look if things appear easier in comparison to conventional algebra.

If you recognise what you have as vector rotation, you can write down the answer without any calculations!

The equation you derived shows that the magnitude of a vector is invariant under rotation.

 

[Soumalya]

3

If you recognise what you have as vector rotation, you can write down the answer without any calculations!

The equation you derived shows that the magnitude of a vector is invariant under rotation.

I apologize for using boldface type alphabets but the equation is a simple transcendental equation I used bold alphabets to highlight the quantities primarily involved in my problem.Noting that it's a simple scalar equation how can I solve for a and b in terms of c₁, c₂ and θ?

 

[zoki85]

I got stuck while solving a problem in statics unable to solve a system of equations of the form obtained as below,

a cosθ - b sinθ = c₁
a sinθ + b cosθ = c₂

I wish to obtain a and b in terms of c₁ and c₂.Squaring and adding I obtained,

a² + b² = c₁² + c₂²

I need another relation in a,b,c₁ and c₂ to achieve the result.How do I proceed next?

These are not transcedental equations. If just Θ is unknown It is not even system of equations since you have only one indepedent variable in each equation. Such trigonometric equations can be solved by transforming sum of two trigonometric function in one trigonometric function. See method http://www.sinclair.edu/centers/mathlab/pub/findyourcourse/worksheets/DiffEq/ConvertingTheForm_asinx.pdf .
Or one can use theory of complex numbers.

 

[PeroK]

I apologize for using boldface type alphabets but the equation is a simple transcendental equation I used bold alphabets to highlight the quantities primarily involved in my problem.Noting that it's a simple scalar equation how can I solve for a and b in terms of c₁, c₂ and θ?

Why not

$a = c_1 cos(\theta) + c_2 sin(\theta)$
$b = -c_1 sin(\theta) + c_2 cos(\theta)$

 

[Soumalya]

Why not

$a = c_1 cos(\theta) + c_2 sin(\theta)$
$b = -c_1 sin(\theta) + c_2 cos(\theta)$

Yes!

Thank You :)

 

[Soumalya]

Why not

$a = c_1 cos(\theta) + c_2 sin(\theta)$
$b = -c_1 sin(\theta) + c_2 cos(\theta)$

Could you briefly outline the procedure that was used to obtain the solution?

 

[PeroK]

Could you briefly outline the procedure that was used to obtain the solution?

I just wrote it down.

$(c_1, c_2)$ is the vector obtained by rotating the vector $(a, b)$ by $-\theta$ about the origin.

So, to obtain $(a, b)$ you rotate $(c_1, c2)$ by $+\theta$

 

[Soumalya]

I just wrote it down.

$(c_1, c_2)$ is the vector obtained by rotating the vector $(a, b)$ by $-\theta$ about the origin.

So, to obtain $(a, b)$ you rotate $(c_1, c2)$ by $+\theta$

Is there an easy analytical way to arrive to the solution using ordinary algebra rather than using linear algebra?

 

[PeroK]

Is there an easy analytical way to arrive to the solution using ordinary algebra rather than using linear algebra?

Multiply the first equation by $cos\theta$ and the second by $sin\theta$ and add to get $a$.

 

[Soumalya]

Thank You once again :)

 

[Mark44]

I got stuck while solving a problem in statics unable to solve a system of equations of the form obtained as below,

a cosθ - b sinθ = c₁
a sinθ + b cosθ = c₂

I wish to obtain a and b in terms of c₁ and c₂.Squaring and adding I obtained,

a² + b² = c₁² + c₂²

I need another relation in a,b,c₁ and c₂ to achieve the result.How do I proceed next?

In future posts, please do not decorate what you write with all of the font features. This is not permitted under PF rules (https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/). What I have emphasized below applies to specific items in your post.

Fonts:
When posting a new topic do not use the CAPS lock (all-CAPS), bold, oversized, non-standard, or brightly colored fonts, or any combination thereof. They are hard to read and are considered yelling. When replying in an existing topic it is fine to use CAPS or bold to highlight main points.

 

[Soumalya]

In future posts, please do not decorate what you write with all of the font features. This is not permitted under PF rules (https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/). What I have emphasized below applies to specific items in your post.

I would take care of that in future :)

 

[HallsofIvy]

I got stuck while solving a problem in statics unable to solve a system of equations of the form obtained as below,

a cosθ - b sinθ = c₁
a sinθ + b cosθ = c₂

I wish to obtain a and b in terms of c₁ and c₂.Squaring and adding I obtained,

a² + b² = c₁² + c₂²

I need another relation in a,b,c₁ and c₂ to achieve the result.How do I proceed next?

Multiply the first equation by cos(\theta): a cos^2(\theta)- b cos(\theta)sin(\theta)= c_1 cos(\theta)
Multiply the second equation by sin(\theta): a sin^2(\theta)+ b sin(\theta)cos(\theta)= c_2 sin(\theta)
Add those two equations: a= c_1 cos(\theta)a+ c_2 sin(\theta).

Multiply the first equation by sin(\theta), the second by cos(\theta), and subtract to get b.