Господин Экзамен
sin(pi-t)*cos(2*pi-t)/tan(pi-t)*cos(pi-t) если t=2
Решение
Вы ввели
[src]
sin(pi - t)*cos(2*pi - t)*cos(pi - t)
-------------------------------------
tan(pi - t)
$$\frac{\sin{\left(- t + \pi \right)} \cos{\left(- t + \pi \right)} \cos{\left(- t + 2 \pi \right)}}{\tan{\left(- t + \pi \right)}}$$
sin(pi - t)*cos(2*pi - t)*cos(pi - t)/tan(pi - t)
Подстановка условия
[src]
sin(pi - t)*cos(2*pi - t)*cos(pi - t)/tan(pi - t) при t = 2
подставляем
sin(pi - t)*cos(2*pi - t)*cos(pi - t)
-------------------------------------
tan(pi - t)
$$\frac{\sin{\left(- t + \pi \right)} \cos{\left(- t + \pi \right)} \cos{\left(- t + 2 \pi \right)}}{\tan{\left(- t + \pi \right)}}$$
3 cos (t)
$$\cos^{3}{\left(t \right)}$$
переменные
t = 2
$$t = 2$$
3 cos ((2))
$$\cos^{3}{\left((2) \right)}$$
3 cos (2)
$$\cos^{3}{\left(2 \right)}$$
cos(2)^3
Собрать выражение
[src]
cos(3*t) 3*cos(t) -------- + -------- 4 4
$$\frac{3 \cos{\left(t \right)}}{4} + \frac{\cos{\left(3 t \right)}}{4}$$
cos(3*t)/4 + 3*cos(t)/4
Степени
[src]
2
cos (t)*sin(t)
--------------
tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
/ I*(pi - t) I*(t - pi)\ / I*(t - 2*pi) I*(-t + 2*pi)\
|e e | |e e | / I*(t - pi) I*(pi - t)\ / I*(pi - t) I*(t - pi)\
-|----------- + -----------|*|------------- + --------------|*\- e + e /*\e + e /
\ 2 2 / \ 2 2 /
------------------------------------------------------------------------------------------------------------------------
/ I*(pi - t) I*(t - pi)\
2*\- e + e /
$$- \frac{\left(\frac{e^{i \left(- t + \pi\right)}}{2} + \frac{e^{i \left(t - \pi\right)}}{2}\right) \left(e^{i \left(- t + \pi\right)} - e^{i \left(t - \pi\right)}\right) \left(e^{i \left(- t + \pi\right)} + e^{i \left(t - \pi\right)}\right) \left(\frac{e^{i \left(- t + 2 \pi\right)}}{2} + \frac{e^{i \left(t - 2 \pi\right)}}{2}\right)}{2 \left(- e^{i \left(- t + \pi\right)} + e^{i \left(t - \pi\right)}\right)}$$
-(exp(i*(pi - t))/2 + exp(i*(t - pi))/2)*(exp(i*(t - 2*pi))/2 + exp(i*(-t + 2*pi))/2)*(-exp(i*(t - pi)) + exp(i*(pi - t)))*(exp(i*(pi - t)) + exp(i*(t - pi)))/(2*(-exp(i*(pi - t)) + exp(i*(t - pi))))
Численный ответ
[src]
cos(pi - t)*cos(2*pi - t)*sin(pi - t)/tan(pi - t)
cos(pi - t)*cos(2*pi - t)*sin(pi - t)/tan(pi - t)
Объединение рациональных выражений
[src]
2
cos (t)*sin(t)
--------------
tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
cos(t)^2*sin(t)/tan(t)
Комбинаторика
[src]
2
cos (t)*sin(t)
--------------
tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
cos(t)^2*sin(t)/tan(t)
Тригонометрическая часть
[src]
3 cos (t)
$$\cos^{3}{\left(t \right)}$$
1 ------- 3 sec (t)
$$\frac{1}{\sec^{3}{\left(t \right)}}$$
3/ pi\
sin |t + --|
\ 2 /
$$\sin^{3}{\left(t + \frac{\pi}{2} \right)}$$
1
------------
3/pi \
csc |-- - t|
\2 /
$$\frac{1}{\csc^{3}{\left(- t + \frac{\pi}{2} \right)}}$$
3 / 2/t\\ -|-1 + 2*sin |-|| \ \2//
$$- \left(2 \sin^{2}{\left(\frac{t}{2} \right)} - 1\right)^{3}$$
(1 + cos(2*t))*cos(t)
---------------------
2
$$\frac{\left(\cos{\left(2 t \right)} + 1\right) \cos{\left(t \right)}}{2}$$
2
cos (t)*sin(t)
--------------
tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
csc(t)
------------------
2
2*csc(2*t)*sec (t)
$$\frac{\csc{\left(t \right)}}{2 \csc{\left(2 t \right)} \sec^{2}{\left(t \right)}}$$
2
cos (t)*sin(2*t)
----------------
2*sin(t)
$$\frac{\sin{\left(2 t \right)} \cos^{2}{\left(t \right)}}{2 \sin{\left(t \right)}}$$
3 / 2/t\\ 6/t\ |-1 + cot |-|| *sin |-| \ \2// \2/
$$\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{3} \sin^{6}{\left(\frac{t}{2} \right)}$$
3 / 2/t\\ 6/t\ |1 - tan |-|| *cos |-| \ \2// \2/
$$\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{3} \cos^{6}{\left(\frac{t}{2} \right)}$$
2/ pi\
sin |t + --|*sin(2*t)
\ 2 /
---------------------
2*sin(t)
$$\frac{\sin{\left(2 t \right)} \sin^{2}{\left(t + \frac{\pi}{2} \right)}}{2 \sin{\left(t \right)}}$$
/1 cos(2*t)\
|- + --------|*sin(t)
\2 2 /
---------------------
tan(t)
$$\frac{\left(\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right) \sin{\left(t \right)}}{\tan{\left(t \right)}}$$
2 / pi\
cos (t)*cos|2*t - --|
\ 2 /
---------------------
/ pi\
2*cos|t - --|
\ 2 /
$$\frac{\cos^{2}{\left(t \right)} \cos{\left(2 t - \frac{\pi}{2} \right)}}{2 \cos{\left(t - \frac{\pi}{2} \right)}}$$
/ pi\
sec|t - --|
\ 2 /
-----------------------
2 / pi\
2*sec (t)*sec|2*t - --|
\ 2 /
$$\frac{\sec{\left(t - \frac{\pi}{2} \right)}}{2 \sec^{2}{\left(t \right)} \sec{\left(2 t - \frac{\pi}{2} \right)}}$$
3
/ 2/t\\
|-1 + cot |-||
\ \2//
---------------
3
/ 2/t\\
|1 + cot |-||
\ \2//
$$\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{3}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{3}}$$
3
/ 2/t\\
|1 - tan |-||
\ \2//
--------------
3
/ 2/t\\
|1 + tan |-||
\ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{3}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{3}}$$
3 /t\
cos (t)*(1 + cos(t))*tan|-|
\2/
---------------------------
sin(t)
$$\frac{\left(\cos{\left(t \right)} + 1\right) \cos^{3}{\left(t \right)} \tan{\left(\frac{t}{2} \right)}}{\sin{\left(t \right)}}$$
3/t pi\
8*tan |- + --|
\2 4 /
-------------------
3
/ 2/t pi\\
|1 + tan |- + --||
\ \2 4 //
$$\frac{8 \tan^{3}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{3}}$$
2 / 2/t\\ 6/t\ /t\ 2*|1 - tan |-|| *cos |-|*cot(t)*tan|-| \ \2// \2/ \2/
$$2 \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos^{6}{\left(\frac{t}{2} \right)} \tan{\left(\frac{t}{2} \right)} \cot{\left(t \right)}$$
/ 2 2 \
|1 cos (t) sin (t)|
|- + ------- - -------|*sin(t)
\2 2 2 /
------------------------------
tan(t)
$$\frac{\left(- \frac{\sin^{2}{\left(t \right)}}{2} + \frac{\cos^{2}{\left(t \right)}}{2} + \frac{1}{2}\right) \sin{\left(t \right)}}{\tan{\left(t \right)}}$$
2
/ 2/t\\ 6/t\ /t\
2*|1 - tan |-|| *cos |-|*tan|-|
\ \2// \2/ \2/
-------------------------------
tan(t)
$$\frac{2 \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos^{6}{\left(\frac{t}{2} \right)} \tan{\left(\frac{t}{2} \right)}}{\tan{\left(t \right)}}$$
2 2 / /t\\ / /t\\ 4/t\ |1 + tan|-|| *|-1 + tan|-|| *cos |-|*cos(t) \ \2// \ \2// \2/
$$\left(\tan{\left(\frac{t}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{t}{2} \right)} \cos{\left(t \right)}$$
/ 1 for t mod 2*pi = 0
|
| 3
$$\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{3} \sin^{6}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}$$
2
/ 2/t\\ /t\
2*|1 - tan |-|| *tan|-|
\ \2// \2/
-----------------------
3
/ 2/t\\
|1 + tan |-|| *tan(t)
\ \2//
$$\frac{2 \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{3} \tan{\left(t \right)}}$$
2 2
2 / /t\\ / /t\\
(1 + cos(t)) *|1 + tan|-|| *|-1 + tan|-|| *cos(t)
\ \2// \ \2//
-------------------------------------------------
4
$$\frac{\left(\cos{\left(t \right)} + 1\right)^{2} \left(\tan{\left(\frac{t}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos{\left(t \right)}}{4}$$
2
/ 1 \
2*|1 - -------| *cot(t)
| 2/t\|
| cot |-||
\ \2//
-----------------------
3
/ 1 \ /t\
|1 + -------| *cot|-|
| 2/t\| \2/
| cot |-||
\ \2//
$$\frac{2 \left(1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cot{\left(t \right)}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{3} \cot{\left(\frac{t}{2} \right)}}$$
/ 1 for t mod 2*pi = 0 | | 3 |/ 2/t\\ ||-1 + cot |-|| <\ \2// |--------------- otherwise | 3 | / 2/t\\ | |1 + cot |-|| \ \ \2//
$$\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{3}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{3}} & \text{otherwise} \end{cases}$$
2 6
/ 2/t\\ / 2/t\\ 12/t\ /t\
2*|1 - tan |-|| *|1 - tan |-|| *cos |-|*tan|-|
\ \2// \ \4// \4/ \2/
-----------------------------------------------
tan(t)
$$\frac{2 \left(- \tan^{2}{\left(\frac{t}{4} \right)} + 1\right)^{6} \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos^{12}{\left(\frac{t}{4} \right)} \tan{\left(\frac{t}{2} \right)}}{\tan{\left(t \right)}}$$
2 6
/ 2/t\\ / 2/t\\ /t\
2*|1 - tan |-|| *|1 - tan |-|| *tan|-|
\ \2// \ \4// \2/
--------------------------------------
6
/ 2/t\\
|1 + tan |-|| *tan(t)
\ \4//
$$\frac{2 \left(- \tan^{2}{\left(\frac{t}{4} \right)} + 1\right)^{6} \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{4} \right)} + 1\right)^{6} \tan{\left(t \right)}}$$
2
/ 4/t\\
| 4*sin |-||
| \2/| 2/t\ 6/pi t\
2*|1 - ---------| *sin |-|*sin |-- + -|*sin(2*t)
| 2 | \2/ \2 2/
\ sin (t) /
------------------------------------------------
3
sin (t)
$$\frac{2 \left(- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(\frac{t}{2} \right)} \sin{\left(2 t \right)} \sin^{6}{\left(\frac{t}{2} + \frac{\pi}{2} \right)}}{\sin^{3}{\left(t \right)}}$$
// 1 for t mod 2*pi = 0\
// 0 for t mod pi = 0\ || |
|< |*|< 2 |*cot(t)
\\sin(t) otherwise / ||cos (t) otherwise |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}$$
// 1 for t mod 2*pi = 0\
// 0 for t mod pi = 0\ || |
|< |*|< 2 |
\\sin(t) otherwise / ||cos (t) otherwise |
\\ /
----------------------------------------------------------
tan(t)
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)}{\tan{\left(t \right)}}$$
2
/ 2/t pi\\
| cos |- - --||
| \2 2 /| 5/t\ /t pi\
2*|1 - ------------| *cos |-|*cos(t)*cos|- - --|
| 2/t\ | \2/ \2 2 /
| cos |-| |
\ \2/ /
------------------------------------------------
/ pi\
cos|t - --|
\ 2 /
$$\frac{2 \left(1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{5}{\left(\frac{t}{2} \right)} \cos{\left(t \right)} \cos{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos{\left(t - \frac{\pi}{2} \right)}}$$
2
/ 2/t\ \
| sec |-| |
| \2/ | / pi\
2*|1 - ------------| *sec|t - --|
| 2/t pi\| \ 2 /
| sec |- - --||
\ \2 2 //
---------------------------------
5/t\ /t pi\
sec(t)*sec |-|*sec|- - --|
\2/ \2 2 /
$$\frac{2 \left(- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(t - \frac{\pi}{2} \right)}}{\sec^{5}{\left(\frac{t}{2} \right)} \sec{\left(t \right)} \sec{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/|
2*|1 - ------------| *csc(t)
| 2/t\ |
| csc |-| |
\ \2/ /
-------------------------------
/t\ /pi \ 5/pi t\
csc|-|*csc|-- - t|*csc |-- - -|
\2/ \2 / \2 2/
$$\frac{2 \left(1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc{\left(t \right)}}{\csc{\left(\frac{t}{2} \right)} \csc{\left(- t + \frac{\pi}{2} \right)} \csc^{5}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}$$
2
/ 4/t\\
| 4*sin |-||
| \2/| 2/t\
2*|1 - ---------| *sin |-|*sin(2*t)
| 2 | \2/
\ sin (t) /
-----------------------------------
3
/ 4/t\\
| 4*sin |-||
| \2/| 3
|1 + ---------| *sin (t)
| 2 |
\ sin (t) /
$$\frac{2 \left(- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(\frac{t}{2} \right)} \sin{\left(2 t \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{3} \sin^{3}{\left(t \right)}}$$
2/t pi\ / 2/t\\
2*tan |- + --|*|1 + tan |-||*tan(t)
\2 4 / \ \2//
----------------------------------------
2
/ 2 \ / 2/t pi\\ /t\
\1 + tan (t)/*|1 + tan |- + --|| *tan|-|
\ \2 4 // \2/
$$\frac{2 \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right) \tan{\left(t \right)} \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(t \right)} + 1\right) \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \tan{\left(\frac{t}{2} \right)}}$$
// 1 for t mod 2*pi = 0\
// 0 for t mod pi = 0\ || |
|< |*|< 2/ pi\ |*sin(2*t)
\\sin(t) otherwise / ||sin |t + --| otherwise |
\\ \ 2 / /
------------------------------------------------------------------------
2
2*sin (t)
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\sin^{2}{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \sin{\left(2 t \right)}}{2 \sin^{2}{\left(t \right)}}$$
// 0 for t mod pi = 0\
|| | // 1 for t mod 2*pi = 0\
||1 - cos(t) | || |
|<---------- otherwise |*|< 2 |
|| /t\ | ||cos (t) otherwise |
|| tan|-| | \\ /
\\ \2/ /
--------------------------------------------------------------
tan(t)
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{- \cos{\left(t \right)} + 1}{\tan{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)}{\tan{\left(t \right)}}$$
// t \
|| 1 for - mod 2*pi = 0|
2 || 2 |
/ 2/t\\ || | /t\
2*|1 - tan |-|| *|< 6 |*cot(t)*tan|-|
\ \2// ||/ 2/t\\ 12/t\ | \2/
|||-1 + cot |-|| *sin |-| otherwise |
||\ \4// \4/ |
\\ /
$$2 \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{t}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{t}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{t}{4} \right)} & \text{otherwise} \end{cases}\right) \tan{\left(\frac{t}{2} \right)} \cot{\left(t \right)}$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|< / pi\ |*|< 2 |*cos(t)
||cos|t - --| otherwise | ||cos (t) otherwise |
\\ \ 2 / / \\ /
----------------------------------------------------------------------
/ pi\
cos|t - --|
\ 2 /
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\cos{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) \cos{\left(t \right)}}{\cos{\left(t - \frac{\pi}{2} \right)}}$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 1 | || 1 | / pi\
|<----------- otherwise |*|<------- otherwise |*sec|t - --|
|| / pi\ | || 2 | \ 2 /
||sec|t - --| | ||sec (t) |
\\ \ 2 / / \\ /
---------------------------------------------------------------------------
sec(t)
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sec{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) \sec{\left(t - \frac{\pi}{2} \right)}}{\sec{\left(t \right)}}$$
// 1 for t mod 2*pi = 0\
// 0 for t mod pi = 0\ || |
|| | || 1 |
|< 1 |*|<------------ otherwise |*csc(t)
||------ otherwise | || 2/pi \ |
\\csc(t) / ||csc |-- - t| |
\\ \2 / /
----------------------------------------------------------------------
/pi \
csc|-- - t|
\2 /
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\csc{\left(t \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \csc{\left(t \right)}}{\csc{\left(- t + \frac{\pi}{2} \right)}}$$
// t \
|| 1 for - mod 2*pi = 0|
|| 2 |
|| |
2 || 6 |
/ 1 \ ||/ 2/t\\ |
2*|1 - -------| *|<|-1 + cot |-|| |*cot(t)
| 2/t\| ||\ \4// |
| cot |-|| ||--------------- otherwise |
\ \2// || 6 |
|| / 2/t\\ |
|| |1 + cot |-|| |
\\ \ \4// /
--------------------------------------------------------------
/t\
cot|-|
\2/
$$\frac{2 \left(1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{t}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{t}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}}{\cot{\left(\frac{t}{2} \right)}}$$
2
/ 2/t pi\\
| cos |- - --||
| \2 2 /| /t pi\
2*|1 - ------------| *cos(t)*cos|- - --|
| 2/t\ | \2 2 /
| cos |-| |
\ \2/ /
----------------------------------------
3
/ 2/t pi\\
| cos |- - --||
| \2 2 /| /t\ / pi\
|1 + ------------| *cos|-|*cos|t - --|
| 2/t\ | \2/ \ 2 /
| cos |-| |
\ \2/ /
$$\frac{2 \left(1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos{\left(t \right)} \cos{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{3} \cos{\left(\frac{t}{2} \right)} \cos{\left(t - \frac{\pi}{2} \right)}}$$
2
/ 2/t\ \
| sec |-| |
| \2/ | /t\ / pi\
2*|1 - ------------| *sec|-|*sec|t - --|
| 2/t pi\| \2/ \ 2 /
| sec |- - --||
\ \2 2 //
----------------------------------------
3
/ 2/t\ \
| sec |-| |
| \2/ | /t pi\
|1 + ------------| *sec(t)*sec|- - --|
| 2/t pi\| \2 2 /
| sec |- - --||
\ \2 2 //
$$\frac{2 \left(- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(\frac{t}{2} \right)} \sec{\left(t - \frac{\pi}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{3} \sec{\left(t \right)} \sec{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/| /pi t\
2*|1 - ------------| *csc(t)*csc|-- - -|
| 2/t\ | \2 2/
| csc |-| |
\ \2/ /
----------------------------------------
3
/ 2/pi t\\
| csc |-- - -||
| \2 2/| /t\ /pi \
|1 + ------------| *csc|-|*csc|-- - t|
| 2/t\ | \2/ \2 /
| csc |-| |
\ \2/ /
$$\frac{2 \left(1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc{\left(t \right)} \csc{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{3} \csc{\left(\frac{t}{2} \right)} \csc{\left(- t + \frac{\pi}{2} \right)}}$$
// 1 for t mod 2*pi = 0\
// 0 for t mod pi = 0\ || |
|| | || 2 |
|| /t\ | ||/ 2/t\\ |
|| 2*cot|-| | |||-1 + cot |-|| |
|< \2/ |*|<\ \2// |*cot(t)
||----------- otherwise | ||--------------- otherwise |
|| 2/t\ | || 2 |
||1 + cot |-| | || / 2/t\\ |
\\ \2/ / || |1 + cot |-|| |
\\ \ \2// /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}$$
// 1 for t mod 2*pi = 0\
// 0 for t mod pi = 0\ || |
|| | ||/ 1 for t mod 2*pi = 0 |
|
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}$$
// 1 for t mod 2*pi = 0\
// 0 for t mod pi = 0\ || |
|| | || 2 |
|| /t\ | ||/ 2/t\\ |
|| 2*tan|-| | |||1 - tan |-|| |
|< \2/ |*|<\ \2// |
||----------- otherwise | ||-------------- otherwise |
|| 2/t\ | || 2 |
||1 + tan |-| | ||/ 2/t\\ |
\\ \2/ / |||1 + tan |-|| |
\\\ \2// /
----------------------------------------------------------------------
tan(t)
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{\tan{\left(t \right)}}$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
// 0 for t mod pi = 0\ ||/ 1 \ |
|| | |||-1 + -------| |
|| 2 | ||| 2/t\| |
||-------------------- otherwise | ||| tan |-|| |
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right) \tan{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)}{\tan{\left(t \right)}}$$
// / pi\ \
|| 0 for |t + --| mod pi = 0| // zoo for t mod pi = 0\
// 0 for 2*t mod pi = 0\ || \ 2 / | || |
|< |*|< |*|< 1 |
\\sin(2*t) otherwise / || 2 2/t pi\ | ||------ otherwise |
||(1 + sin(t)) *cot |- + --| otherwise | \\sin(t) /
\\ \2 4 / /
------------------------------------------------------------------------------------------------------------------
2
$$\frac{\left(\begin{cases} 0 & \text{for}\: 2 t \bmod \pi = 0 \\\sin{\left(2 t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(t \right)} + 1\right)^{2} \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin{\left(t \right)}} & \text{otherwise} \end{cases}\right)}{2}$$
// 1 for t mod 2*pi = 0\
// 0 for t mod pi = 0\ || |
|| | ||/ 1 for t mod 2*pi = 0 |
||/ 0 for t mod pi = 0 | ||| |
||| | ||| 2 |
||| /t\ | |||/ 2/t\\ |
|<| 2*cot|-| |*|<||-1 + cot |-|| |*cot(t)
||< \2/ otherwise | ||<\ \2// otherwise |
|||----------- otherwise | |||--------------- otherwise |
||| 2/t\ | ||| 2 |
|||1 + cot |-| | ||| / 2/t\\ |
\\\ \2/ / ||| |1 + cot |-|| |
\\\ \ \2// /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
// 0 for t mod pi = 0\ ||/ 2 \ |
|| | ||| sin (t) | |
|| sin(t) | |||-1 + ---------| |
||----------------------- otherwise | ||| 4/t\| |
||/ 2 \ | ||| 4*sin |-|| |
|<| sin (t) | 2/t\ |*|<\ \2// |*sin(2*t)
|||1 + ---------|*sin |-| | ||----------------- otherwise |
||| 4/t\| \2/ | || 2 |
||| 4*sin |-|| | || / 2 \ |
||\ \2// | || | sin (t) | |
\\ / || |1 + ---------| |
|| | 4/t\| |
|| | 4*sin |-|| |
\\ \ \2// /
----------------------------------------------------------------------------------------------
2
2*sin (t)
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right) \sin^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) \sin{\left(2 t \right)}}{2 \sin^{2}{\left(t \right)}}$$
// 1 for t mod 2*pi = 0\
|| |
// 0 for t mod pi = 0\ || 2 |
|| | ||/1 2 cos(2*t)\ |
|| 2*(-sin(2*t) + 2*sin(t)) | |||- - (1 - cos(t)) - --------| |
|<------------------------------ otherwise |*|<\2 2 / |
|| 2 | ||------------------------------- otherwise |
||1 - cos(2*t) + 2*(1 - cos(t)) | || 2 |
\\ / ||/1 2 cos(2*t)\ |
|||- + (1 - cos(t)) - --------| |
\\\2 2 / /
----------------------------------------------------------------------------------------------------------
tan(t)
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cdot \left(2 \sin{\left(t \right)} - \sin{\left(2 t \right)}\right)}{2 \left(- \cos{\left(t \right)} + 1\right)^{2} - \cos{\left(2 t \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- \left(- \cos{\left(t \right)} + 1\right)^{2} - \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)^{2}}{\left(\left(- \cos{\left(t \right)} + 1\right)^{2} - \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)^{2}} & \text{otherwise} \end{cases}\right)}{\tan{\left(t \right)}}$$
// / pi\ \
|| 0 for |t + --| mod pi = 0|
|| \ 2 / | // zoo for t mod pi = 0\
// 0 for 2*t mod pi = 0\ || | || |
|| | || 2/t pi\ | || 2/t\ |
|| 2*cot(t) | || 4*cot |- + --| | ||1 + cot |-| |
|<----------- otherwise |*|< \2 4 / |*|< \2/ |
|| 2 | ||------------------- otherwise | ||----------- otherwise |
||1 + cot (t) | || 2 | || /t\ |
\\ / ||/ 2/t pi\\ | || 2*cot|-| |
|||1 + cot |- + --|| | \\ \2/ /
||\ \2 4 // |
\\ /
-------------------------------------------------------------------------------------------------------------------
2
$$\frac{\left(\begin{cases} 0 & \text{for}\: 2 t \bmod \pi = 0 \\\frac{2 \cot{\left(t \right)}}{\cot^{2}{\left(t \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} + 1}{2 \cot{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)}{2}$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
// 0 for t mod pi = 0\ ||/ 2/t\ \ |
|| | ||| cos |-| | |
|| /t\ | ||| \2/ | |
|| 2*cos|-| | |||-1 + ------------| |
|| \2/ | ||| 2/t pi\| |
||------------------------------ otherwise | ||| cos |- - --|| |
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{t}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \cos{\left(t \right)}}{\cos{\left(t - \frac{\pi}{2} \right)}}$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
// 0 for t mod pi = 0\ ||/ 2/t pi\\ |
|| | ||| sec |- - --|| |
|| /t pi\ | ||| \2 2 /| |
|| 2*sec|- - --| | |||-1 + ------------| |
|| \2 2 / | ||| 2/t\ | |
||------------------------- otherwise | ||| sec |-| | | / pi\
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right) \sec{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) \sec{\left(t - \frac{\pi}{2} \right)}}{\sec{\left(t \right)}}$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
// 0 for t mod pi = 0\ ||/ 2/t\ \ |
|| | ||| csc |-| | |
|| /t\ | ||| \2/ | |
|| 2*csc|-| | |||-1 + ------------| |
|| \2/ | ||| 2/pi t\| |
||------------------------------ otherwise | ||| csc |-- - -|| |
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{t}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \csc{\left(t \right)}}{\csc{\left(- t + \frac{\pi}{2} \right)}}$$
Piecewise((0, Mod(t = pi, 0)), (2*csc(t/2)/((1 + csc(t/2)^2/csc(pi/2 - t/2)^2)*csc(pi/2 - t/2)), True))*Piecewise((1, Mod(t = 2*pi, 0)), ((-1 + csc(t/2)^2/csc(pi/2 - t/2)^2)^2/(1 + csc(t/2)^2/csc(pi/2 - t/2)^2)^2, True))*csc(t)/csc(pi/2 - t)
Рациональный знаменатель
[src]
2
cos (t)*sin(t)
--------------
tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
cos(t)^2*sin(t)/tan(t)
Общий знаменатель
[src]
2
cos (t)*sin(t)
--------------
tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
cos(t)^2*sin(t)/tan(t)