Mister Exam
Expression simplification/
Least common denominator/
cos(sqrt(g/h)*t-sqrt((l+h)/h)*acos((d*(h+l))/(l*(2*z+d))))
Least common denominator cos(sqrt(g/h)*t-sqrt((l+h)/h)*acos((d*(h+l))/(l*(2*z+d))))
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//
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)))
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)))))
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)))
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
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)))
Trigonometric part
[src]
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
|t* / - / 1 + - *acos|---------||
2| \/ h \/ h \ d + 2*z /|
-1 + cot |--------- - ---------------------------|
\ 2 2 /
--------------------------------------------------
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
|t* / - / 1 + - *acos|---------||
2| \/ h \/ h \ d + 2*z /|
1 + cot |--------- - ---------------------------|
\ 2 2 /
$$\frac{\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)} - 1}{\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)} + 1}$$
1 --------------------------------------------------- / _______ ___\ |pi / h + l / d*(h + l) \ / g | csc|-- + / ----- *acos|-----------| - t* / - | \2 \/ h \l*(d + 2*z)/ \/ h /
$$\frac{1}{\csc{\left(- t \sqrt{\frac{g}{h}} + \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} + \frac{\pi}{2} \right)}}$$
1 ---------------------------------------------- / ___ _______ \ | / g / h + l / d*(h + l) \| sec|t* / - - / ----- *acos|-----------|| \ \/ h \/ h \l*(d + 2*z)//
$$\frac{1}{\sec{\left(t \sqrt{\frac{g}{h}} - \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} \right)}}$$
/ _______ ___\
| / h + l / d*(h + l) \ / g |
| / ----- *acos|-----------| t* / - |
2|pi \/ h \l*(d + 2*z)/ \/ h |
csc |-- + ----------------------------- - ---------|
\2 2 2 /
1 - ----------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
csc |--------- - -----------------------------|
\ 2 2 /
--------------------------------------------------------
/ _______ ___\
| / h + l / d*(h + l) \ / g |
| / ----- *acos|-----------| t* / - |
2|pi \/ h \l*(d + 2*z)/ \/ h |
csc |-- + ----------------------------- - ---------|
\2 2 2 /
1 + ----------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
csc |--------- - -----------------------------|
\ 2 2 /
$$\frac{1 - \frac{\csc^{2}{\left(- \frac{t \sqrt{\frac{g}{h}}}{2} + \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{t \sqrt{\frac{g}{h}}}{2} + \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}}$$
1 ------------------------------------------------- / / / h\\ \ | _______ |d*|1 + -|| ___| |pi / l | \ l/| / g | csc|-- + / 1 + - *acos|---------| - t* / - | \2 \/ h \ d + 2*z / \/ h /
$$\frac{1}{\csc{\left(- t \sqrt{\frac{g}{h}} + \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)} + \frac{\pi}{2} \right)}}$$
/ ___ _______ \ |pi / g / h + l / d*(h + l) \| sin|-- + t* / - - / ----- *acos|-----------|| \2 \/ h \/ h \l*(d + 2*z)//
$$\sin{\left(t \sqrt{\frac{g}{h}} - \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} + \frac{\pi}{2} \right)}$$
/ ___ _______ \
| / g / h + l / d*(h + l) \|
| t* / - / ----- *acos|-----------||
2| pi \/ h \/ h \l*(d + 2*z)/|
cos |- -- + --------- - -----------------------------|
\ 2 2 2 /
1 - ------------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
cos |--------- - -----------------------------|
\ 2 2 /
----------------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
| t* / - / ----- *acos|-----------||
2| pi \/ h \/ h \l*(d + 2*z)/|
cos |- -- + --------- - -----------------------------|
\ 2 2 2 /
1 + ------------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
cos |--------- - -----------------------------|
\ 2 2 /
$$\frac{1 - \frac{\cos^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}}$$
/ / / / h\\\\
| | ___ _______ |d*|1 + -||||
| | / g / l | \ l/||| / / / h\\\
cos|2*|t* / - - / 1 + - *acos|---------||| | ___ _______ |d*|1 + -|||
1 \ \ \/ h \/ h \ d + 2*z /// | / g / l | \ l/||
- - - ------------------------------------------------ + cos|t* / - - / 1 + - *acos|---------||
2 2 \ \/ h \/ h \ d + 2*z //
-----------------------------------------------------------------------------------------------------
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
1 - cos|t* / - - / 1 + - *acos|---------||
\ \/ h \/ h \ d + 2*z //
$$\frac{\cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)} \right)} - \frac{\cos{\left(2 \left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}\right) \right)}}{2} - \frac{1}{2}}{1 - \cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)} \right)}}$$
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
4| \/ h \/ h \l*(d + 2*z)/|
4*sin |--------- - -----------------------------|
\ 2 2 /
1 - -------------------------------------------------
/ ___ _______ \
2| / g / h + l / d*(h + l) \|
sin |t* / - - / ----- *acos|-----------||
\ \/ h \/ h \l*(d + 2*z)//
-----------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
4| \/ h \/ h \l*(d + 2*z)/|
4*sin |--------- - -----------------------------|
\ 2 2 /
1 + -------------------------------------------------
/ ___ _______ \
2| / g / h + l / d*(h + l) \|
sin |t* / - - / ----- *acos|-----------||
\ \/ h \/ h \l*(d + 2*z)//
$$\frac{- \frac{4 \sin^{4}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}{\sin^{2}{\left(t \sqrt{\frac{g}{h}} - \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}{\sin^{2}{\left(t \sqrt{\frac{g}{h}} - \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} \right)}} + 1}$$
/ / / h\\ \
| _______ |d*|1 + -|| ___|
| / l | \ l/| / g |
| / 1 + - *acos|---------| t* / - |
2| \/ h \ d + 2*z / \/ h | / 1 \
4*tan |- --------------------------- + ---------|*|-1 + -----------------------------------------------|
\ 4 4 / | / ___ _______ \|
| | / g / h + l / d*(h + l) \||
| |t* / - / ----- *acos|-----------|||
| 2| \/ h \/ h \l*(d + 2*z)/||
| tan |--------- - -----------------------------||
\ \ 2 2 //
--------------------------------------------------------------------------------------------------------
2
/ / / / h\\ \\
| | _______ |d*|1 + -|| ___||
| | / l | \ l/| / g ||
| | / 1 + - *acos|---------| t* / - ||
| 2| \/ h \ d + 2*z / \/ h ||
|1 + tan |- --------------------------- + ---------||
\ \ 4 4 //
$$\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}\right) \tan^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{4} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{4} \right)}}{\left(\tan^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{4} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{4} \right)} + 1\right)^{2}}$$
1
1 - -----------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
cot |--------- - -----------------------------|
\ 2 2 /
---------------------------------------------------
1
1 + -----------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
cot |--------- - -----------------------------|
\ 2 2 /
$$\frac{1 - \frac{1}{\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}}$$
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
csc |--------- - -----------------------------|
\ 2 2 /
-1 + ----------------------------------------------------
/ _______ ___\
| / h + l / d*(h + l) \ / g |
| / ----- *acos|-----------| t* / - |
2|pi \/ h \l*(d + 2*z)/ \/ h |
csc |-- + ----------------------------- - ---------|
\2 2 2 /
---------------------------------------------------------
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
|t* / - / 1 + - *acos|---------||
2| \/ h \/ h \ d + 2*z /|
csc |--------- - ---------------------------|
\ 2 2 /
$$\frac{\frac{\csc^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}{\csc^{2}{\left(- \frac{t \sqrt{\frac{g}{h}}}{2} + \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} + \frac{\pi}{2} \right)}} - 1}{\csc^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)}}$$
/ / / h\\ \ / / / h\\\
| _______ |d*|1 + -|| ___| | ___ _______ |d*|1 + -||| / / ___ _______ \\
| / l | \ l/| / g | | / g / l | \ l/|| | | / g / h + l / d*(h + l) \||
| / 1 + - *acos|---------| t* / - | |t* / - - / 1 + - *acos|---------|| | |t* / - / ----- *acos|-----------|||
2| \/ h \ d + 2*z / \/ h | 4| \/ h \/ h \ d + 2*z /| | 2| \/ h \/ h \l*(d + 2*z)/||
4*cot |- --------------------------- + ---------|*sin |---------------------------------------|*|-1 + cot |--------- - -----------------------------||
\ 4 4 / \ 4 / \ \ 2 2 //
$$4 \left(\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)} - 1\right) \sin^{4}{\left(\frac{t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{4} \right)} \cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{4} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{4} \right)}$$
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
1 - tan |--------- - -----------------------------|
\ 2 2 /
---------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
1 + tan |--------- - -----------------------------|
\ 2 2 /
$$\frac{1 - \tan^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}{\tan^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)} + 1}$$
/ / ___ _______ \ \
/ / / h\\\ | | / g / h + l / d*(h + l) \| |
| ___ _______ |d*|1 + -||| | |t* / - / ----- *acos|-----------|| |
| / g / l | \ l/|| | 2| \/ h \/ h \l*(d + 2*z)/| |
| t* / - / 1 + - *acos|---------|| | cos |--------- - -----------------------------| |
2| pi \/ h \/ h \ d + 2*z /| | \ 2 2 / |
cos |- -- + --------- - ---------------------------|*|-1 + ------------------------------------------------------|
\ 2 2 2 / | / ___ _______ \|
| | / g / h + l / d*(h + l) \||
| | t* / - / ----- *acos|-----------|||
| 2| pi \/ h \/ h \l*(d + 2*z)/||
| cos |- -- + --------- - -----------------------------||
\ \ 2 2 2 //
$$\left(\frac{\cos^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}{\cos^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} - \frac{\pi}{2} \right)}$$
/ / / h\\ \
| _______ |d*|1 + -|| ___| / / ___ _______ \\
| / l | \ l/| / g | | | / g / h + l / d*(h + l) \||
| / 1 + - *acos|---------| t* / - | | |t* / - / ----- *acos|-----------|||
2| \/ h \ d + 2*z / \/ h | | 2| \/ h \/ h \l*(d + 2*z)/||
4*cot |- --------------------------- + ---------|*|-1 + cot |--------- - -----------------------------||
\ 4 4 / \ \ 2 2 //
--------------------------------------------------------------------------------------------------------
2
/ / / / h\\ \\
| | _______ |d*|1 + -|| ___||
| | / l | \ l/| / g ||
| | / 1 + - *acos|---------| t* / - ||
| 2| \/ h \ d + 2*z / \/ h ||
|1 + cot |- --------------------------- + ---------||
\ \ 4 4 //
$$\frac{4 \left(\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)} - 1\right) \cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{4} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{4} \right)}}{\left(\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{4} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{4} \right)} + 1\right)^{2}}$$
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
|t* / - - / 1 + - *acos|---------||
4| \/ h \/ h \ d + 2*z /|
4*sin |---------------------------------------|
\ 2 /
1 - -----------------------------------------------
/ ___ _______ \
2| / g / h + l / d*(h + l) \|
sin |t* / - - / ----- *acos|-----------||
\ \/ h \/ h \l*(d + 2*z)//
---------------------------------------------------
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
|t* / - - / 1 + - *acos|---------||
4| \/ h \/ h \ d + 2*z /|
4*sin |---------------------------------------|
\ 2 /
1 + -----------------------------------------------
/ ___ _______ \
2| / g / h + l / d*(h + l) \|
sin |t* / - - / ----- *acos|-----------||
\ \/ h \/ h \l*(d + 2*z)//
$$\frac{1 - \frac{4 \sin^{4}{\left(\frac{t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)}}{\sin^{2}{\left(t \sqrt{\frac{g}{h}} - \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} \right)}}}{1 + \frac{4 \sin^{4}{\left(\frac{t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)}}{\sin^{2}{\left(t \sqrt{\frac{g}{h}} - \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} \right)}}}$$
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
-1 + cot |--------- - -----------------------------|
\ 2 2 /
----------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
1 + cot |--------- - -----------------------------|
\ 2 2 /
$$\frac{\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)} - 1}{\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)} + 1}$$
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
|t* / - - / 1 + - *acos|---------||
2| \/ h \/ h \ d + 2*z /| / 1 \
cos |---------------------------------------|*|1 - -----------------------------------------------|
\ 2 / | / ___ _______ \|
| | / g / h + l / d*(h + l) \||
| |t* / - / ----- *acos|-----------|||
| 2| \/ h \/ h \l*(d + 2*z)/||
| cot |--------- - -----------------------------||
\ \ 2 2 //
$$\left(1 - \frac{1}{\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}\right) \cos^{2}{\left(\frac{t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)}$$
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/|| / / / h\\\
|t* / - - / 1 + - *acos|---------|| | ___ _______ |d*|1 + -|||
2| \/ h \/ h \ d + 2*z /| | / g / l | \ l/||
-2*sin |---------------------------------------|*cos|t* / - - / 1 + - *acos|---------||
\ 2 / \ \/ h \/ h \ d + 2*z //
---------------------------------------------------------------------------------------------
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
-1 + cos|t* / - - / 1 + - *acos|---------||
\ \/ h \/ h \ d + 2*z //
$$- \frac{2 \sin^{2}{\left(\frac{t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)} \cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)} \right)}}{\cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)} \right)} - 1}$$
/ / / h\\\
| ___ _______ |d*|1 + -||| / / ___ _______ \ \
| / g / l | \ l/|| | 2| / g / h + l / d*(h + l) \| |
|t* / - / 1 + - *acos|---------|| | sin |t* / - - / ----- *acos|-----------|| |
2| \/ h \/ h \ d + 2*z /| | \ \/ h \/ h \l*(d + 2*z)// |
sin |--------- - ---------------------------|*|-1 + -------------------------------------------------|
\ 2 2 / | / ___ _______ \|
| | / g / h + l / d*(h + l) \||
| |t* / - / ----- *acos|-----------|||
| 4| \/ h \/ h \l*(d + 2*z)/||
| 4*sin |--------- - -----------------------------||
\ \ 2 2 //
$$\left(-1 + \frac{\sin^{2}{\left(t \sqrt{\frac{g}{h}} - \sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)} \right)}}{4 \sin^{4}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}\right) \sin^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)}$$
/ / / h\\\ | ___ _______ |d*|1 + -||| |pi / g / l | \ l/|| sin|-- + t* / - - / 1 + - *acos|---------|| \2 \/ h \/ h \ d + 2*z //
$$\sin{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)} + \frac{\pi}{2} \right)}$$
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
|t* / - / 1 + - *acos|---------||
2| \/ h \/ h \ d + 2*z /|
1 - tan |--------- - ---------------------------|
\ 2 2 /
-------------------------------------------------
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
|t* / - / 1 + - *acos|---------||
2| \/ h \/ h \ d + 2*z /|
1 + tan |--------- - ---------------------------|
\ 2 2 /
$$\frac{1 - \tan^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)}}{\tan^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)} + 1}$$
/ / / h\\\ | ___ _______ |d*|1 + -||| | / g / l | \ l/|| cos|t* / - - / 1 + - *acos|---------|| \ \/ h \/ h \ d + 2*z //
$$\cos{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)} \right)}$$
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
sec |--------- - -----------------------------|
\ 2 2 /
1 - ------------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
| t* / - / ----- *acos|-----------||
2| pi \/ h \/ h \l*(d + 2*z)/|
sec |- -- + --------- - -----------------------------|
\ 2 2 2 /
----------------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
sec |--------- - -----------------------------|
\ 2 2 /
1 + ------------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
| t* / - / ----- *acos|-----------||
2| pi \/ h \/ h \l*(d + 2*z)/|
sec |- -- + --------- - -----------------------------|
\ 2 2 2 /
$$\frac{- \frac{\sec^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}{\sec^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}{\sec^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} - \frac{\pi}{2} \right)}} + 1}$$
1 -------------------------------------------- / / / h\\\ | ___ _______ |d*|1 + -||| | / g / l | \ l/|| sec|t* / - - / 1 + - *acos|---------|| \ \/ h \/ h \ d + 2*z //
$$\frac{1}{\sec{\left(t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)} \right)}}$$
/ ___ _______ \
| / g / h + l / d*(h + l) \|
| t* / - / ----- *acos|-----------||
2| pi \/ h \/ h \l*(d + 2*z)/|
sec |- -- + --------- - -----------------------------|
\ 2 2 2 /
-1 + ------------------------------------------------------
/ ___ _______ \
| / g / h + l / d*(h + l) \|
|t* / - / ----- *acos|-----------||
2| \/ h \/ h \l*(d + 2*z)/|
sec |--------- - -----------------------------|
\ 2 2 /
-----------------------------------------------------------
/ / / h\\\
| ___ _______ |d*|1 + -|||
| / g / l | \ l/||
| t* / - / 1 + - *acos|---------||
2| pi \/ h \/ h \ d + 2*z /|
sec |- -- + --------- - ---------------------------|
\ 2 2 2 /
$$\frac{-1 + \frac{\sec^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)}}}{\sec^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} - \frac{\pi}{2} \right)}}$$
/ / / h\\\
| ___ _______ |d*|1 + -||| / / ___ _______ \\
| / g / l | \ l/|| | | / g / h + l / d*(h + l) \||
|t* / - - / 1 + - *acos|---------|| | |t* / - / ----- *acos|-----------|||
2| \/ h \/ h \ d + 2*z /| | 2| \/ h \/ h \l*(d + 2*z)/||
sin |---------------------------------------|*|-1 + cot |--------- - -----------------------------||
\ 2 / \ \ 2 2 //
$$\left(\cot^{2}{\left(\frac{t \sqrt{\frac{g}{h}}}{2} - \frac{\sqrt{\frac{h + l}{h}} \operatorname{acos}{\left(\frac{d \left(h + l\right)}{l \left(d + 2 z\right)} \right)}}{2} \right)} - 1\right) \sin^{2}{\left(\frac{t \sqrt{\frac{g}{h}} - \sqrt{1 + \frac{l}{h}} \operatorname{acos}{\left(\frac{d \left(\frac{h}{l} + 1\right)}{d + 2 z} \right)}}{2} \right)}$$
sin((t*sqrt(g/h) - sqrt(1 + l/h)*acos(d*(1 + h/l)/(d + 2*z)))/2)^2*(-1 + cot(t*sqrt(g/h)/2 - sqrt((h + l)/h)*acos(d*(h + l)/(l*(d + 2*z)))/2)^2)