Solve for X as shown in the below sketch

[henneh]

Homework Statement

As shown in the attached sketch, derive a non-transcendental equation for the angle, X, given that the parameters A,B, d and R are known variables.

Homework Equations

Basic trig.

The Attempt at a Solution

I have derived an equation which is a function of X as shown below based on simple trignometry:

B = d + R + Rcos(x) + dcos(x) + (A-d-R)sin(x)

Which can be arranged into a (somewhat) easier format:

(R+d)cos(x) + (A-d-R)sin(x) = B - d - R

However I cannot simplify this expression any further. If anyone can help me with this I would really appreciate it! I hope I have been clear in showing the problem geometry, any ambiguities please let me know. Thank you so much.

 

[mfb]

Every sum of A*cos(x)+B*sin(x) can be combined to a single function like C*sin(x+D). Those identities help to calculate C and D.

 

[SammyS]

Homework Statement

As shown in the attached sketch, derive a non-transcendental equation for the angle, X, given that the parameters A,B, d and R are known variables.

Homework Equations

Basic trig.

The Attempt at a Solution

I have derived an equation which is a function of X as shown below based on simple trigonometry:

B = d + R + Rcos(x) + dcos(x) + (A-d-R)sin(x)

Which can be arranged into a (somewhat) easier format:

(R+d)cos(x) + (A-d-R)sin(x) = B - d - R

However I cannot simplify this expression any further. If anyone can help me with this I would really appreciate it! I hope I have been clear in showing the problem geometry, any ambiguities please let me know. Thank you so much.

Look at the "Linear Combination" section of "List of trigonometric identities" in Wikipedia by using the following link.

http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinations

Essentially, $\displaystyle A\sin x+B\cos x=\sqrt{A^2+B^{\,2}}\cdot\sin(x+\varphi)$

where $\displaystyle\ \varphi = \arctan \left(\frac{B}{A}\right) + \begin{cases} 0 & \text{if , }A \ge 0, \\ \pi & \text{if , }A \lt 0, \end{cases}$

(I see mfb beat me to it !)

 

[henneh]

Thanks guys for the help! : )