Господин Экзамен
Общий знаменатель sin(3*x)/6-sin(9*x)/18
Решение
Вы ввели
[src]
sin(3*x) sin(9*x) -------- - -------- 6 18
$$\frac{\sin{\left(3 x \right)}}{6} - \frac{\sin{\left(9 x \right)}}{18}$$
sin(3*x)/6 - sin(9*x)/18
Численный ответ
[src]
0.166666666666667*sin(3*x) - 0.0555555555555556*sin(9*x)
0.166666666666667*sin(3*x) - 0.0555555555555556*sin(9*x)
Объединение рациональных выражений
[src]
-sin(9*x) + 3*sin(3*x)
----------------------
18
$$\frac{3 \sin{\left(3 x \right)} - \sin{\left(9 x \right)}}{18}$$
(-sin(9*x) + 3*sin(3*x))/18
Рациональный знаменатель
[src]
-6*sin(9*x) + 18*sin(3*x)
-------------------------
108
$$\frac{18 \sin{\left(3 x \right)} - 6 \sin{\left(9 x \right)}}{108}$$
(-6*sin(9*x) + 18*sin(3*x))/108
Степени
[src]
/ -3*I*x 3*I*x\ / -9*I*x 9*I*x\
I*\- e + e / I*\- e + e /
- ---------------------- + ----------------------
12 36
$$- \frac{i \left(e^{3 i x} - e^{- 3 i x}\right)}{12} + \frac{i \left(e^{9 i x} - e^{- 9 i x}\right)}{36}$$
-i*(-exp(-3*i*x) + exp(3*i*x))/12 + i*(-exp(-9*i*x) + exp(9*i*x))/36
Тригонометрическая часть
[src]
1 1 - ----------- + ---------- 18*csc(9*x) 6*csc(3*x)
$$- \frac{1}{18 \csc{\left(9 x \right)}} + \frac{1}{6 \csc{\left(3 x \right)}}$$
1 1 - ---------------- + --------------- 18*csc(pi - 9*x) 6*csc(pi - 3*x)
$$\frac{1}{6 \csc{\left(- 3 x + \pi \right)}} - \frac{1}{18 \csc{\left(- 9 x + \pi \right)}}$$
/ pi\ / pi\
cos|9*x - --| cos|3*x - --|
\ 2 / \ 2 /
- ------------- + -------------
18 6
$$\frac{\cos{\left(3 x - \frac{\pi}{2} \right)}}{6} - \frac{\cos{\left(9 x - \frac{\pi}{2} \right)}}{18}$$
1 1
- ---------------- + ---------------
/ pi\ / pi\
18*sec|9*x - --| 6*sec|3*x - --|
\ 2 / \ 2 /
$$- \frac{1}{18 \sec{\left(9 x - \frac{\pi}{2} \right)}} + \frac{1}{6 \sec{\left(3 x - \frac{\pi}{2} \right)}}$$
1 1
- ---------------- + ---------------
/pi \ /pi \
18*sec|-- - 9*x| 6*sec|-- - 3*x|
\2 / \2 /
$$\frac{1}{6 \sec{\left(- 3 x + \frac{\pi}{2} \right)}} - \frac{1}{18 \sec{\left(- 9 x + \frac{\pi}{2} \right)}}$$
/9*x\ /3*x\
(1 + cos(9*x))*tan|---| (1 + cos(3*x))*tan|---|
\ 2 / \ 2 /
- ----------------------- + -----------------------
18 6
$$\frac{\left(\cos{\left(3 x \right)} + 1\right) \tan{\left(\frac{3 x}{2} \right)}}{6} - \frac{\left(\cos{\left(9 x \right)} + 1\right) \tan{\left(\frac{9 x}{2} \right)}}{18}$$
/9*x\ /3*x\
cot|---| cot|---|
\ 2 / \ 2 /
- ----------------- + -----------------
/ 2/9*x\\ / 2/3*x\\
9*|1 + cot |---|| 3*|1 + cot |---||
\ \ 2 // \ \ 2 //
$$- \frac{\cot{\left(\frac{9 x}{2} \right)}}{9 \left(\cot^{2}{\left(\frac{9 x}{2} \right)} + 1\right)} + \frac{\cot{\left(\frac{3 x}{2} \right)}}{3 \left(\cot^{2}{\left(\frac{3 x}{2} \right)} + 1\right)}$$
/9*x\ /3*x\
tan|---| tan|---|
\ 2 / \ 2 /
- ----------------- + -----------------
/ 2/9*x\\ / 2/3*x\\
9*|1 + tan |---|| 3*|1 + tan |---||
\ \ 2 // \ \ 2 //
$$- \frac{\tan{\left(\frac{9 x}{2} \right)}}{9 \left(\tan^{2}{\left(\frac{9 x}{2} \right)} + 1\right)} + \frac{\tan{\left(\frac{3 x}{2} \right)}}{3 \left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right)}$$
1 1
- -------------------------- + --------------------------
/ 1 \ /9*x\ / 1 \ /3*x\
9*|1 + ---------|*cot|---| 3*|1 + ---------|*cot|---|
| 2/9*x\| \ 2 / | 2/3*x\| \ 2 /
| cot |---|| | cot |---||
\ \ 2 // \ \ 2 //
$$- \frac{1}{9 \cdot \left(1 + \frac{1}{\cot^{2}{\left(\frac{9 x}{2} \right)}}\right) \cot{\left(\frac{9 x}{2} \right)}} + \frac{1}{3 \cdot \left(1 + \frac{1}{\cot^{2}{\left(\frac{3 x}{2} \right)}}\right) \cot{\left(\frac{3 x}{2} \right)}}$$
/ 2/pi 9*x\\ / 2/pi 3*x\\
|1 - cot |-- + ---||*(1 + sin(9*x)) |1 - cot |-- + ---||*(1 + sin(3*x))
\ \4 2 // \ \4 2 //
- ----------------------------------- + -----------------------------------
36 12
$$\frac{\left(- \cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(3 x \right)} + 1\right)}{12} - \frac{\left(- \cot^{2}{\left(\frac{9 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(9 x \right)} + 1\right)}{36}$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
< <
\sin(9*x) otherwise \sin(3*x) otherwise
- ----------------------------- + -----------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\sin{\left(9 x \right)} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
< 1 < 1
|-------- otherwise |-------- otherwise
\csc(9*x) \csc(3*x)
- ----------------------------- + -----------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{1}{\csc{\left(3 x \right)}} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{1}{\csc{\left(9 x \right)}} & \text{otherwise} \end{cases}}{18}\right)$$
2/pi 9*x\ 2/pi 3*x\
-1 + tan |-- + ---| -1 + tan |-- + ---|
\4 2 / \4 2 /
- ----------------------- + ----------------------
/ 2/pi 9*x\\ / 2/pi 3*x\\
18*|1 + tan |-- + ---|| 6*|1 + tan |-- + ---||
\ \4 2 // \ \4 2 //
$$\frac{\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} - 1}{6 \left(\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{\tan^{2}{\left(\frac{9 x}{2} + \frac{\pi}{4} \right)} - 1}{18 \left(\tan^{2}{\left(\frac{9 x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
2/pi 9*x\ 2/pi 3*x\
1 - cot |-- + ---| 1 - cot |-- + ---|
\4 2 / \4 2 /
- ----------------------- + ----------------------
/ 2/pi 9*x\\ / 2/pi 3*x\\
18*|1 + cot |-- + ---|| 6*|1 + cot |-- + ---||
\ \4 2 // \ \4 2 //
$$\frac{- \cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1}{6 \left(\cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{- \cot^{2}{\left(\frac{9 x}{2} + \frac{\pi}{4} \right)} + 1}{18 \left(\cot^{2}{\left(\frac{9 x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
< / pi\ < / pi\
|cos|9*x - --| otherwise |cos|3*x - --| otherwise
\ \ 2 / \ \ 2 /
- ---------------------------------- + ----------------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\cos{\left(3 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\cos{\left(9 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
| 1 | 1
<------------- otherwise <------------- otherwise
| / pi\ | / pi\
|sec|9*x - --| |sec|3*x - --|
\ \ 2 / \ \ 2 /
- ---------------------------------- + ----------------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{1}{\sec{\left(9 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{18}\right)$$
/ / 3*pi\ / / 3*pi\
| 1 for |9*x + ----| mod 2*pi = 0 | 1 for |3*x + ----| mod 2*pi = 0
< \ 2 / < \ 2 /
| |
\sin(9*x) otherwise \sin(3*x) otherwise
- ---------------------------------------- + ----------------------------------------
18 6
$$\left(\frac{\begin{cases} 1 & \text{for}\: \left(3 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \left(9 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(9 x \right)} & \text{otherwise} \end{cases}}{18}\right)$$
2/9*x\ 2/3*x\
2*sin |---|*sin(9*x) 2*sin |---|*sin(3*x)
\ 2 / \ 2 /
- --------------------------- + ---------------------------
/ 2 4/9*x\\ / 2 4/3*x\\
9*|sin (9*x) + 4*sin |---|| 3*|sin (3*x) + 4*sin |---||
\ \ 2 // \ \ 2 //
$$- \frac{2 \sin^{2}{\left(\frac{9 x}{2} \right)} \sin{\left(9 x \right)}}{9 \cdot \left(4 \sin^{4}{\left(\frac{9 x}{2} \right)} + \sin^{2}{\left(9 x \right)}\right)} + \frac{2 \sin^{2}{\left(\frac{3 x}{2} \right)} \sin{\left(3 x \right)}}{3 \cdot \left(4 \sin^{4}{\left(\frac{3 x}{2} \right)} + \sin^{2}{\left(3 x \right)}\right)}$$
2/9*x\ 2/3*x\
2*sin |---| 2*sin |---|
\ 2 / \ 2 /
- ---------------------------- + ----------------------------
/ 4/9*x\\ / 4/3*x\\
| 4*sin |---|| | 4*sin |---||
| \ 2 /| | \ 2 /|
9*|1 + -----------|*sin(9*x) 3*|1 + -----------|*sin(3*x)
| 2 | | 2 |
\ sin (9*x) / \ sin (3*x) /
$$- \frac{2 \sin^{2}{\left(\frac{9 x}{2} \right)}}{9 \cdot \left(\frac{4 \sin^{4}{\left(\frac{9 x}{2} \right)}}{\sin^{2}{\left(9 x \right)}} + 1\right) \sin{\left(9 x \right)}} + \frac{2 \sin^{2}{\left(\frac{3 x}{2} \right)}}{3 \cdot \left(\frac{4 \sin^{4}{\left(\frac{3 x}{2} \right)}}{\sin^{2}{\left(3 x \right)}} + 1\right) \sin{\left(3 x \right)}}$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
|1 - cos(9*x) |1 - cos(3*x)
<------------ otherwise <------------ otherwise
| /9*x\ | /3*x\
| tan|---| | tan|---|
\ \ 2 / \ \ 2 /
- --------------------------------- + ---------------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{- \cos{\left(3 x \right)} + 1}{\tan{\left(\frac{3 x}{2} \right)}} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{- \cos{\left(9 x \right)} + 1}{\tan{\left(\frac{9 x}{2} \right)}} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
| /9*x\ | /3*x\
| 2*cot|---| | 2*cot|---|
< \ 2 / < \ 2 /
|------------- otherwise |------------- otherwise
| 2/9*x\ | 2/3*x\
|1 + cot |---| |1 + cot |---|
\ \ 2 / \ \ 2 /
- ---------------------------------- + ----------------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{9 x}{2} \right)}}{\cot^{2}{\left(\frac{9 x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
| /9*x\ | /3*x\
| 2*tan|---| | 2*tan|---|
< \ 2 / < \ 2 /
|------------- otherwise |------------- otherwise
| 2/9*x\ | 2/3*x\
|1 + tan |---| |1 + tan |---|
\ \ 2 / \ \ 2 /
- ---------------------------------- + ----------------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{9 x}{2} \right)}}{\tan^{2}{\left(\frac{9 x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\sin{\left(9 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{18}\right)$$
/9*x\ /3*x\
sec|---| sec|---|
\ 2 / \ 2 /
- ---------------------------------------- + ----------------------------------------
/ 2/9*x\ \ / 2/3*x\ \
| sec |---| | | sec |---| |
| \ 2 / | / pi 9*x\ | \ 2 / | / pi 3*x\
9*|1 + ----------------|*sec|- -- + ---| 3*|1 + ----------------|*sec|- -- + ---|
| 2/ pi 9*x\| \ 2 2 / | 2/ pi 3*x\| \ 2 2 /
| sec |- -- + ---|| | sec |- -- + ---||
\ \ 2 2 // \ \ 2 2 //
$$- \frac{\sec{\left(\frac{9 x}{2} \right)}}{9 \left(\frac{\sec^{2}{\left(\frac{9 x}{2} \right)}}{\sec^{2}{\left(\frac{9 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{9 x}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(\frac{3 x}{2} \right)}}{3 \left(\frac{\sec^{2}{\left(\frac{3 x}{2} \right)}}{\sec^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
| 2 | 2
|------------------------ otherwise |------------------------ otherwise
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{3 x}{2} \right)}}\right) \tan{\left(\frac{3 x}{2} \right)}} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{9 x}{2} \right)}}\right) \tan{\left(\frac{9 x}{2} \right)}} & \text{otherwise} \end{cases}}{18}\right)$$
/ pi 9*x\ / pi 3*x\
cos|- -- + ---| cos|- -- + ---|
\ 2 2 / \ 2 2 /
- --------------------------------- + ---------------------------------
/ 2/ pi 9*x\\ / 2/ pi 3*x\\
| cos |- -- + ---|| | cos |- -- + ---||
| \ 2 2 /| /9*x\ | \ 2 2 /| /3*x\
9*|1 + ----------------|*cos|---| 3*|1 + ----------------|*cos|---|
| 2/9*x\ | \ 2 / | 2/3*x\ | \ 2 /
| cos |---| | | cos |---| |
\ \ 2 / / \ \ 2 / /
$$- \frac{\cos{\left(\frac{9 x}{2} - \frac{\pi}{2} \right)}}{9 \cdot \left(1 + \frac{\cos^{2}{\left(\frac{9 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{9 x}{2} \right)}}\right) \cos{\left(\frac{9 x}{2} \right)}} + \frac{\cos{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{3 \cdot \left(1 + \frac{\cos^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 x}{2} \right)}}\right) \cos{\left(\frac{3 x}{2} \right)}}$$
/pi 9*x\ /pi 3*x\
csc|-- - ---| csc|-- - ---|
\2 2 / \2 2 /
- ------------------------------- + -------------------------------
/ 2/pi 9*x\\ / 2/pi 3*x\\
| csc |-- - ---|| | csc |-- - ---||
| \2 2 /| /9*x\ | \2 2 /| /3*x\
9*|1 + --------------|*csc|---| 3*|1 + --------------|*csc|---|
| 2/9*x\ | \ 2 / | 2/3*x\ | \ 2 /
| csc |---| | | csc |---| |
\ \ 2 / / \ \ 2 / /
$$- \frac{\csc{\left(- \frac{9 x}{2} + \frac{\pi}{2} \right)}}{9 \cdot \left(1 + \frac{\csc^{2}{\left(- \frac{9 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{9 x}{2} \right)}}\right) \csc{\left(\frac{9 x}{2} \right)}} + \frac{\csc{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}}{3 \cdot \left(1 + \frac{\csc^{2}{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 x}{2} \right)}}\right) \csc{\left(\frac{3 x}{2} \right)}}$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
| -2*sin(18*x) + 4*sin(9*x) | -2*sin(6*x) + 4*sin(3*x)
<--------------------------------- otherwise <-------------------------------- otherwise
| 2 | 2
|1 - cos(18*x) + 2*(1 - cos(9*x)) |1 - cos(6*x) + 2*(1 - cos(3*x))
\ \
- ------------------------------------------------------ + -----------------------------------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{4 \sin{\left(3 x \right)} - 2 \sin{\left(6 x \right)}}{2 \left(- \cos{\left(3 x \right)} + 1\right)^{2} - \cos{\left(6 x \right)} + 1} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{4 \sin{\left(9 x \right)} - 2 \sin{\left(18 x \right)}}{2 \left(- \cos{\left(9 x \right)} + 1\right)^{2} - \cos{\left(18 x \right)} + 1} & \text{otherwise} \end{cases}}{18}\right)$$
/ / 3*pi\ / / 3*pi\
| 1 for |9*x + ----| mod 2*pi = 0 | 1 for |3*x + ----| mod 2*pi = 0
| \ 2 / | \ 2 /
| |
| 2/pi 9*x\ | 2/pi 3*x\
<-1 + tan |-- + ---| <-1 + tan |-- + ---|
| \4 2 / | \4 2 /
|------------------- otherwise |------------------- otherwise
| 2/pi 9*x\ | 2/pi 3*x\
| 1 + tan |-- + ---| | 1 + tan |-- + ---|
\ \4 2 / \ \4 2 /
- --------------------------------------------------- + ---------------------------------------------------
18 6
$$\left(\frac{\begin{cases} 1 & \text{for}\: \left(3 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \left(9 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{9 x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{9 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
| sin(9*x) | sin(3*x)
|--------------------------- otherwise |--------------------------- otherwise
|/ 2 \ |/ 2 \
<| sin (9*x) | 2/9*x\ <| sin (3*x) | 2/3*x\
||1 + -----------|*sin |---| ||1 + -----------|*sin |---|
|| 4/9*x\| \ 2 / || 4/3*x\| \ 2 /
|| 4*sin |---|| || 4*sin |---||
|\ \ 2 // |\ \ 2 //
\ \
- ------------------------------------------------ + ------------------------------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{\sin{\left(3 x \right)}}{\left(1 + \frac{\sin^{2}{\left(3 x \right)}}{4 \sin^{4}{\left(\frac{3 x}{2} \right)}}\right) \sin^{2}{\left(\frac{3 x}{2} \right)}} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{\sin{\left(9 x \right)}}{\left(1 + \frac{\sin^{2}{\left(9 x \right)}}{4 \sin^{4}{\left(\frac{9 x}{2} \right)}}\right) \sin^{2}{\left(\frac{9 x}{2} \right)}} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
|/ 0 for 9*x mod pi = 0 |/ 0 for 3*x mod pi = 0
|| ||
|| /9*x\ || /3*x\
<| 2*cot|---| <| 2*cot|---|
|< \ 2 / otherwise |< \ 2 / otherwise
||------------- otherwise ||------------- otherwise
|| 2/9*x\ || 2/3*x\
||1 + cot |---| ||1 + cot |---|
\\ \ 2 / \\ \ 2 /
- ------------------------------------------------------- + -------------------------------------------------------
18 6
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{9 x}{2} \right)}}{\cot^{2}{\left(\frac{9 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
| / pi 9*x\ | / pi 3*x\
| 2*sec|- -- + ---| | 2*sec|- -- + ---|
| \ 2 2 / | \ 2 2 /
|------------------------------- otherwise |------------------------------- otherwise
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 x}{2} \right)}}\right) \sec{\left(\frac{3 x}{2} \right)}} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{9 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{9 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{9 x}{2} \right)}}\right) \sec{\left(\frac{9 x}{2} \right)}} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
| /9*x\ | /3*x\
| 2*cos|---| | 2*cos|---|
| \ 2 / | \ 2 /
|-------------------------------------- otherwise |-------------------------------------- otherwise
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{3 x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{3 x}{2} \right)}}{\cos^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{9 x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{9 x}{2} \right)}}{\cos^{2}{\left(\frac{9 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{9 x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{18}\right)$$
/ 0 for 9*x mod pi = 0 / 0 for 3*x mod pi = 0
| |
| /9*x\ | /3*x\
| 2*csc|---| | 2*csc|---|
| \ 2 / | \ 2 /
|---------------------------------- otherwise |---------------------------------- otherwise
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{3 x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{3 x}{2} \right)}}{\csc^{2}{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{6}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: 9 x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{9 x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{9 x}{2} \right)}}{\csc^{2}{\left(- \frac{9 x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{9 x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{18}\right)$$
-Piecewise((0, Mod(9*x = pi, 0)), (2*csc(9*x/2)/((1 + csc(9*x/2)^2/csc(pi/2 - 9*x/2)^2)*csc(pi/2 - 9*x/2)), True))/18 + Piecewise((0, Mod(3*x = pi, 0)), (2*csc(3*x/2)/((1 + csc(3*x/2)^2/csc(pi/2 - 3*x/2)^2)*csc(pi/2 - 3*x/2)), True))/6