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Expression simplification/ Fraction Decomposition into the simple/ cos(sqrt(g/h)*t-sqrt((l+h)/h)*acos((d*(h+l))/(l*(2*z+d))))

How do you cos(sqrt(g/h)*t-sqrt((l+h)/h)*acos((d*(h+l))/(l*(2*z+d)))) in partial fractions?

An expression to simplify:

The solution

You have entered [src] 
   /    ___         _______                  \
   |   / g         / l + h      / d*(h + l) \|
cos|  /  - *t -   /  ----- *acos|-----------||
   \\/   h      \/     h        \l*(2*z + d)//
$$\cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} \right)}$$ 
cos(sqrt(g/h)*t - sqrt((l + h)/h)*acos((d*(h + l))/((l*(2*z + d)))))
Fraction decomposition [src] 
cos(sqrt(g/h)*t - sqrt(1 + l/h)*acos(d*h/(d*l + 2*l*z) + d*l/(d*l + 2*l*z)))
$$\cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d h}{d l + 2 l z} + \frac{d l}{d l + 2 l z} \right)} \right)}$$ 
   /    ___         _______                                \
   |   / g         /     l      /    d*h           d*l    \|
cos|  /  - *t -   /  1 + - *acos|----------- + -----------||
   \\/   h      \/       h      \d*l + 2*l*z   d*l + 2*l*z//
Numerical answer [src] 
cos(sqrt(g/h)*t - sqrt((l + h)/h)*acos((d*(h + l))/((l*(2*z + d)))))
cos(sqrt(g/h)*t - sqrt((l + h)/h)*acos((d*(h + l))/((l*(2*z + d)))))
Common denominator [src] 
   /      ___       _______                                \
   |     / g       /     l      /    d*h           d*l    \|
cos|t*  /  -  -   /  1 + - *acos|----------- + -----------||
   \  \/   h    \/       h      \d*l + 2*l*z   d*l + 2*l*z//
$$\cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d h}{d l + 2 l z} + \frac{d l}{d l + 2 l z} \right)} \right)}$$ 
cos(t*sqrt(g/h) - sqrt(1 + l/h)*acos(d*h/(d*l + 2*l*z) + d*l/(d*l + 2*l*z)))
Combinatorics [src] 
   /      ___       _______                                \
   |     / g       /     l      /    d*h           d*l    \|
cos|t*  /  -  -   /  1 + - *acos|----------- + -----------||
   \  \/   h    \/       h      \d*l + 2*l*z   d*l + 2*l*z//
$$\cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d h}{d l + 2 l z} + \frac{d l}{d l + 2 l z} \right)} \right)}$$ 
cos(t*sqrt(g/h) - sqrt(1 + l/h)*acos(d*h/(d*l + 2*l*z) + d*l/(d*l + 2*l*z)))
Powers [src] 
   /      ___       _______                  \      /    _______                           ___\
   |     / g       / h + l      / d*(h + l) \|      |   / h + l      / d*(h + l) \        / g |
 I*|t*  /  -  -   /  ----- *acos|-----------||    I*|  /  ----- *acos|-----------| - t*  /  - |
   \  \/   h    \/     h        \l*(d + 2*z)//      \\/     h        \l*(d + 2*z)/     \/   h /
e                                                e                                             
---------------------------------------------- + ----------------------------------------------
                      2                                                2
$$\frac{e^{i \left(- t \sqrt{\frac{g}{h}} + \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}\right)}}{2} + \frac{e^{i \left(t \sqrt{\frac{g}{h}} - \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}\right)}}{2}$$ 
exp(i*(t*sqrt(g/h) - sqrt((h + l)/h)*acos(d*(h + l)/(l*(d + 2*z)))))/2 + exp(i*(sqrt((h + l)/h)*acos(d*(h + l)/(l*(d + 2*z))) - t*sqrt(g/h)))/2
Expand expression [src] 
   /            ___       ___                            \
   |    ___    / 1       / 1    _______     / d*(h + l) \|
cos|t*\/ g *  /  -  -   /  - *\/ l + h *acos|-----------||
   \        \/   h    \/   h                \l*(2*z + d)//
$$\cos{\left(\sqrt{g} t \sqrt{\frac{1}{h}} - \sqrt{h + l} \sqrt{\frac{1}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} \right)}$$ 
   /    ___  \    /    _______                                \      /    ___  \    /    _______                                \
   |   / g   |    |   /     l      /    d*h           d*l    \|      |   / g   |    |   /     l      /    d*h           d*l    \|
cos|  /  - *t|*cos|  /  1 + - *acos|----------- + -----------|| + sin|  /  - *t|*sin|  /  1 + - *acos|----------- + -----------||
   \\/   h   /    \\/       h      \d*l + 2*l*z   d*l + 2*l*z//      \\/   h   /    \\/       h      \d*l + 2*l*z   d*l + 2*l*z//
$$\sin{\left(t \sqrt{\frac{g}{h}} \right)} \sin{\left(\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d h}{d l + 2 l z} + \frac{d l}{d l + 2 l z} \right)} \right)} + \cos{\left(t \sqrt{\frac{g}{h}} \right)} \cos{\left(\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d h}{d l + 2 l z} + \frac{d l}{d l + 2 l z} \right)} \right)}$$ 
cos(sqrt(g/h)*t)*cos(sqrt(1 + l/h)*acos(d*h/(d*l + 2*l*z) + d*l/(d*l + 2*l*z))) + sin(sqrt(g/h)*t)*sin(sqrt(1 + l/h)*acos(d*h/(d*l + 2*l*z) + d*l/(d*l + 2*l*z)))
Rational denominator [src] 
   /      ___       _______                                \
   |     / g       /     l      /    d*h           d*l    \|
cos|t*  /  -  -   /  1 + - *acos|----------- + -----------||
   \  \/   h    \/       h      \d*l + 2*l*z   d*l + 2*l*z//
$$\cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d h}{d l + 2 l z} + \frac{d l}{d l + 2 l z} \right)} \right)}$$ 
cos(t*sqrt(g/h) - sqrt(1 + l/h)*acos(d*h/(d*l + 2*l*z) + d*l/(d*l + 2*l*z)))
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