# Solve for X as shown in the below sketch

1. Jan 3, 2014

### henneh

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

Basic trig.

3. 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.

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2. Jan 3, 2014

### Staff: Mentor

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.

3. Jan 3, 2014

### SammyS

Staff Emeritus
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 !)

4. Jan 4, 2014

### henneh

Thanks guys for the help! : )