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