Господин Экзамен
Общий знаменатель sin(2)*(pi-t)+sin(2)*(pi/2-t)/sin(pi-t)
Решение
Вы ввели
[src]
/pi \
sin(2)*|-- - t|
\2 /
sin(2)*(pi - t) + ---------------
sin(pi - t)
$$\left(- t + \pi\right) \sin{\left(2 \right)} + \frac{\left(- t + \frac{\pi}{2}\right) \sin{\left(2 \right)}}{\sin{\left(- t + \pi \right)}}$$
sin(2)*(pi - t) + sin(2)*(pi/2 - t)/sin(pi - t)
Общее упрощение
[src]
(pi - 2*t + 2*(pi - t)*sin(t))*sin(2)
-------------------------------------
2*sin(t)
$$\frac{\left(2 \left(- t + \pi\right) \sin{\left(t \right)} - 2 t + \pi\right) \sin{\left(2 \right)}}{2 \sin{\left(t \right)}}$$
(pi - 2*t + 2*(pi - t)*sin(t))*sin(2)/(2*sin(t))
Рациональный знаменатель
[src]
pi*sin(2) t*sin(2)
pi*sin(2) - t*sin(2) + --------- - --------
2*sin(t) sin(t)
$$- t \sin{\left(2 \right)} - \frac{t \sin{\left(2 \right)}}{\sin{\left(t \right)}} + \pi \sin{\left(2 \right)} + \frac{\pi \sin{\left(2 \right)}}{2 \sin{\left(t \right)}}$$
pi*sin(2) - 2*t*sin(2) - 2*t*sin(2)*sin(t) + 2*pi*sin(2)*sin(t)
---------------------------------------------------------------
2*sin(t)
$$\frac{- 2 t \sin{\left(2 \right)} \sin{\left(t \right)} - 2 t \sin{\left(2 \right)} + 2 \pi \sin{\left(2 \right)} \sin{\left(t \right)} + \pi \sin{\left(2 \right)}}{2 \sin{\left(t \right)}}$$
(pi*sin(2) - 2*t*sin(2) - 2*t*sin(2)*sin(t) + 2*pi*sin(2)*sin(t))/(2*sin(t))
Собрать выражение
[src]
/ pi*sin(2)\
pi*sin(2) - t*sin(2) - |t*sin(2) - ---------|*csc(t)
\ 2 /
$$- \left(t \sin{\left(2 \right)} - \frac{\pi \sin{\left(2 \right)}}{2}\right) \csc{\left(t \right)} - t \sin{\left(2 \right)} + \pi \sin{\left(2 \right)}$$
pi*sin(2) - t*sin(2) - (t*sin(2) - pi*sin(2)/2)*csc(t)
Общий знаменатель
[src]
pi*sin(2) - 2*t*sin(2)
pi*sin(2) + ---------------------- - t*sin(2)
2*sin(t)
$$- t \sin{\left(2 \right)} + \frac{- 2 t \sin{\left(2 \right)} + \pi \sin{\left(2 \right)}}{2 \sin{\left(t \right)}} + \pi \sin{\left(2 \right)}$$
pi*sin(2) + (pi*sin(2) - 2*t*sin(2))/(2*sin(t)) - t*sin(2)
Объединение рациональных выражений
[src]
(pi - 2*t + 2*(pi - t)*sin(t))*sin(2)
-------------------------------------
2*sin(t)
$$\frac{\left(2 \left(- t + \pi\right) \sin{\left(t \right)} - 2 t + \pi\right) \sin{\left(2 \right)}}{2 \sin{\left(t \right)}}$$
(pi - 2*t + 2*(pi - t)*sin(t))*sin(2)/(2*sin(t))
Комбинаторика
[src]
-(-pi + 2*t - 2*pi*sin(t) + 2*t*sin(t))*sin(2)
-----------------------------------------------
2*sin(t)
$$- \frac{\left(2 t \sin{\left(t \right)} + 2 t - 2 \pi \sin{\left(t \right)} - \pi\right) \sin{\left(2 \right)}}{2 \sin{\left(t \right)}}$$
-(-pi + 2*t - 2*pi*sin(t) + 2*t*sin(t))*sin(2)/(2*sin(t))
Степени
[src]
/pi \
|-- - t|*sin(2)
\2 /
(pi - t)*sin(2) + ---------------
sin(t)
$$\left(- t + \pi\right) \sin{\left(2 \right)} + \frac{\left(- t + \frac{\pi}{2}\right) \sin{\left(2 \right)}}{\sin{\left(t \right)}}$$
/pi \ / -2*I 2*I\ |-- - t|*\- e + e / / -2*I 2*I\ \2 / I*(pi - t)*\- e + e / --------------------------- - --------------------------- I*(t - pi) I*(pi - t) 2 - e + e
$$- \frac{i \left(- t + \pi\right) \left(e^{2 i} - e^{- 2 i}\right)}{2} + \frac{\left(- t + \frac{\pi}{2}\right) \left(e^{2 i} - e^{- 2 i}\right)}{e^{i \left(- t + \pi\right)} - e^{i \left(t - \pi\right)}}$$
(pi/2 - t)*(-exp(-2*i) + exp(2*i))/(-exp(i*(t - pi)) + exp(i*(pi - t))) - i*(pi - t)*(-exp(-2*i) + exp(2*i))/2
Тригонометрическая часть
[src]
pi*csc(t)
pi - t + --------- - t*csc(t)
2
-----------------------------
csc(2)
$$\frac{- t \csc{\left(t \right)} - t + \frac{\pi \csc{\left(t \right)}}{2} + \pi}{\csc{\left(2 \right)}}$$
/ pi*csc(t) \ |pi - t + --------- - t*csc(t)|*sin(2) \ 2 /
$$\left(- t \csc{\left(t \right)} - t + \frac{\pi \csc{\left(t \right)}}{2} + \pi\right) \sin{\left(2 \right)}$$
/ pi t \ |pi - t + -------- - ------|*sin(2) \ 2*sin(t) sin(t)/
$$\left(- t - \frac{t}{\sin{\left(t \right)}} + \pi + \frac{\pi}{2 \sin{\left(t \right)}}\right) \sin{\left(2 \right)}$$
/pi \
(pi - t)*sin(2) + |-- - t|*csc(t)*sin(2)
\2 /
$$\left(- t + \frac{\pi}{2}\right) \sin{\left(2 \right)} \csc{\left(t \right)} + \left(- t + \pi\right) \sin{\left(2 \right)}$$
(pi - 2*t)*sin(2)
(pi - t)*sin(2) + -----------------
2*sin(t)
$$\left(- t + \pi\right) \sin{\left(2 \right)} + \frac{\left(- 2 t + \pi\right) \sin{\left(2 \right)}}{2 \sin{\left(t \right)}}$$
/pi \
|-- - t|*csc(t)
pi - t \2 /
------ + ---------------
csc(2) csc(2)
$$\frac{\left(- t + \frac{\pi}{2}\right) \csc{\left(t \right)}}{\csc{\left(2 \right)}} + \frac{- t + \pi}{\csc{\left(2 \right)}}$$
/ pi\
|-t + --|*sin(2)
\ 2 /
(pi - t)*sin(2) + ----------------
sin(t)
$$\left(- t + \pi\right) \sin{\left(2 \right)} + \frac{\left(- t + \frac{\pi}{2}\right) \sin{\left(2 \right)}}{\sin{\left(t \right)}}$$
/pi \
|-- - t|*sin(2)
\2 /
(pi - t)*sin(2) + ---------------
sin(t)
$$\left(- t + \pi\right) \sin{\left(2 \right)} + \frac{\left(- t + \frac{\pi}{2}\right) \sin{\left(2 \right)}}{\sin{\left(t \right)}}$$
/pi \
|-- - t|*csc(pi - t)
pi - t \2 /
------------ + --------------------
csc(-2 + pi) csc(-2 + pi)
$$\frac{\left(- t + \frac{\pi}{2}\right) \csc{\left(- t + \pi \right)}}{\csc{\left(-2 + \pi \right)}} + \frac{- t + \pi}{\csc{\left(-2 + \pi \right)}}$$
/ pi\
pi*sec|t - --|
\ 2 / / pi\
pi - t + -------------- - t*sec|t - --|
2 \ 2 /
---------------------------------------
/ pi\
sec|2 - --|
\ 2 /
$$\frac{- t \sec{\left(t - \frac{\pi}{2} \right)} - t + \frac{\pi \sec{\left(t - \frac{\pi}{2} \right)}}{2} + \pi}{\sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
/ pi t \
2*|pi - t + -------- - ------|*tan(1)
\ 2*sin(t) sin(t)/
-------------------------------------
2
1 + tan (1)
$$\frac{2 \left(- t - \frac{t}{\sin{\left(t \right)}} + \pi + \frac{\pi}{2 \sin{\left(t \right)}}\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}}$$
/ pi t \ / pi\ |pi - t + ------------- - -----------|*cos|2 - --| | / pi\ / pi\| \ 2 / | 2*cos|t - --| cos|t - --|| \ \ 2 / \ 2 //
$$\left(- t - \frac{t}{\cos{\left(t - \frac{\pi}{2} \right)}} + \pi + \frac{\pi}{2 \cos{\left(t - \frac{\pi}{2} \right)}}\right) \cos{\left(- \frac{\pi}{2} + 2 \right)}$$
(pi - 2*t)*tan(1)
(pi - t)*sin(2) + --------------------
/ 2 \
\1 + tan (1)/*sin(t)
$$\left(- t + \pi\right) \sin{\left(2 \right)} + \frac{\left(- 2 t + \pi\right) \tan{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right) \sin{\left(t \right)}}$$
/pi \ / pi\
|-- - t|*sec|t - --|
pi - t \2 / \ 2 /
----------- + --------------------
/ pi\ / pi\
sec|2 - --| sec|2 - --|
\ 2 / \ 2 /
$$\frac{\left(- t + \frac{\pi}{2}\right) \sec{\left(t - \frac{\pi}{2} \right)}}{\sec{\left(- \frac{\pi}{2} + 2 \right)}} + \frac{- t + \pi}{\sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
/pi \ /pi \
|-- - t|*sec|-- - t|
pi - t \2 / \2 /
------------ + --------------------
/ pi\ / pi\
sec|-2 + --| sec|-2 + --|
\ 2 / \ 2 /
$$\frac{\left(- t + \frac{\pi}{2}\right) \sec{\left(- t + \frac{\pi}{2} \right)}}{\sec{\left(-2 + \frac{\pi}{2} \right)}} + \frac{- t + \pi}{\sec{\left(-2 + \frac{\pi}{2} \right)}}$$
/pi \ /pi \
|-- - t|*sec|-- - t|
pi - t \2 / \2 /
----------- + --------------------
/ pi\ / pi\
sec|2 - --| sec|2 - --|
\ 2 / \ 2 /
$$\frac{\left(- t + \frac{\pi}{2}\right) \sec{\left(- t + \frac{\pi}{2} \right)}}{\sec{\left(- \frac{\pi}{2} + 2 \right)}} + \frac{- t + \pi}{\sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
/pi \ / pi\
|-- - t|*cos|2 - --|
/ pi\ \2 / \ 2 /
(pi - t)*cos|2 - --| + --------------------
\ 2 / / pi\
cos|t - --|
\ 2 /
$$\left(- t + \pi\right) \cos{\left(- \frac{\pi}{2} + 2 \right)} + \frac{\left(- t + \frac{\pi}{2}\right) \cos{\left(- \frac{\pi}{2} + 2 \right)}}{\cos{\left(t - \frac{\pi}{2} \right)}}$$
/pi \
2*|-- - t|*tan(1)
(-2*t + 2*pi)*sin(2) \2 /
-------------------- + --------------------
2 / 2 \
\1 + tan (1)/*sin(t)
$$\frac{\left(- 2 t + 2 \pi\right) \sin{\left(2 \right)}}{2} + \frac{2 \left(- t + \frac{\pi}{2}\right) \tan{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right) \sin{\left(t \right)}}$$
/pi \
|-- - t|*tan(1)
\2 /
(pi - t)*sin(2) + ----------------------------
/ 2 \ 2/t\ /t\
\1 + tan (1)/*cos |-|*tan|-|
\2/ \2/
$$\left(- t + \pi\right) \sin{\left(2 \right)} + \frac{\left(- t + \frac{\pi}{2}\right) \tan{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right) \cos^{2}{\left(\frac{t}{2} \right)} \tan{\left(\frac{t}{2} \right)}}$$
(-4*t + 2*pi)*sin(2)
(pi - t)*sin(2) + ----------------------------
/ 2 \
| 2 sin (2)|
|4*sin (1) + -------|*sin(t)
| 2 |
\ sin (1)/
$$\left(- t + \pi\right) \sin{\left(2 \right)} + \frac{\left(- 4 t + 2 \pi\right) \sin{\left(2 \right)}}{\left(\frac{\sin^{2}{\left(2 \right)}}{\sin^{2}{\left(1 \right)}} + 4 \sin^{2}{\left(1 \right)}\right) \sin{\left(t \right)}}$$
/pi \ // zoo for t mod pi = 0\
(pi - t)*sin(2) + |-- - t|*|< |*sin(2)
\2 / \\csc(t) otherwise /
$$\left(\left(- t + \frac{\pi}{2}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\csc{\left(t \right)} & \text{otherwise} \end{cases}\right) \sin{\left(2 \right)}\right) + \left(- t + \pi\right) \sin{\left(2 \right)}$$
// zoo for t mod pi = 0\
(pi - 2*t)*|< |
-2*t + 2*pi \\csc(t) otherwise /
----------- + --------------------------------------
2*csc(2) 2*csc(2)
$$\left(\frac{\left(- 2 t + \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\csc{\left(t \right)} & \text{otherwise} \end{cases}\right)}{2 \csc{\left(2 \right)}}\right) + \frac{- 2 t + 2 \pi}{2 \csc{\left(2 \right)}}$$
// zoo for t mod pi = 0\
/pi \ || |
(pi - t)*sin(2) + |-- - t|*|< 1 |*sin(2)
\2 / ||------ otherwise |
\\sin(t) /
$$\left(\left(- t + \frac{\pi}{2}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin{\left(t \right)}} & \text{otherwise} \end{cases}\right) \sin{\left(2 \right)}\right) + \left(- t + \pi\right) \sin{\left(2 \right)}$$
2 /pi \
4*sin (1)*|-- - t|
(-2*t + 2*pi)*sin(2) \2 /
-------------------- + -----------------------------
2 / 4 \
| 4*sin (1)|
|1 + ---------|*sin(2)*sin(t)
| 2 |
\ sin (2) /
$$\frac{\left(- 2 t + 2 \pi\right) \sin{\left(2 \right)}}{2} + \frac{4 \left(- t + \frac{\pi}{2}\right) \sin^{2}{\left(1 \right)}}{\left(1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}\right) \sin{\left(2 \right)} \sin{\left(t \right)}}$$
(-2*t + 2*pi)*sin(2) (pi - 2*t)*tan(1)
-------------------- + ----------------------------------------------
2 / 2 2 \ /t\
\1 + tan (1) + tan (1)*cos(t) + cos(t)/*tan|-|
\2/
$$\frac{\left(- 2 t + 2 \pi\right) \sin{\left(2 \right)}}{2} + \frac{\left(- 2 t + \pi\right) \tan{\left(1 \right)}}{\left(\cos{\left(t \right)} + \cos{\left(t \right)} \tan^{2}{\left(1 \right)} + 1 + \tan^{2}{\left(1 \right)}\right) \tan{\left(\frac{t}{2} \right)}}$$
/pi \ / pi\
2*|-- - t|*csc(t)*csc|-1 + --|
-2*t + 2*pi \2 / \ 2 /
----------- + ------------------------------
2*csc(2) / 2/ pi\\
| csc |-1 + --||
| \ 2 /|
|1 + -------------|*csc(1)
| 2 |
\ csc (1) /
$$\frac{2 \left(- t + \frac{\pi}{2}\right) \csc{\left(t \right)} \csc{\left(-1 + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(1 \right)}}\right) \csc{\left(1 \right)}} + \frac{- 2 t + 2 \pi}{2 \csc{\left(2 \right)}}$$
// zoo for t mod pi = 0\
|| |
(pi - 2*t)*|< 1 |*sin(2)
||------ otherwise |
(-2*t + 2*pi)*sin(2) \\sin(t) /
-------------------- + ---------------------------------------------
2 2
$$\left(\frac{\left(- 2 t + \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin{\left(t \right)}} & \text{otherwise} \end{cases}\right) \sin{\left(2 \right)}}{2}\right) + \frac{\left(- 2 t + 2 \pi\right) \sin{\left(2 \right)}}{2}$$
/ 2/t\\ /pi \
|1 + cot |-||*|-- - t|*cot(1)
2*(pi - t)*cot(1) \ \2// \2 /
----------------- + -----------------------------
2 / 2 \ /t\
1 + cot (1) \1 + cot (1)/*cot|-|
\2/
$$\frac{2 \left(- t + \pi\right) \cot{\left(1 \right)}}{\cot^{2}{\left(1 \right)} + 1} + \frac{\left(- t + \frac{\pi}{2}\right) \left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right) \cot{\left(1 \right)}}{\left(\cot^{2}{\left(1 \right)} + 1\right) \cot{\left(\frac{t}{2} \right)}}$$
/ 2/t\\ /pi \
|1 + tan |-||*|-- - t|*tan(1)
2*(pi - t)*tan(1) \ \2// \2 /
----------------- + -----------------------------
2 / 2 \ /t\
1 + tan (1) \1 + tan (1)/*tan|-|
\2/
$$\frac{2 \left(- t + \pi\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}} + \frac{\left(- t + \frac{\pi}{2}\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right) \tan{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right) \tan{\left(\frac{t}{2} \right)}}$$
/ 2/t\\ /pi \
|1 + tan |-||*|-- - t|*tan(1)
(-2*t + 2*pi)*tan(1) \ \2// \2 /
-------------------- + -----------------------------
2 / 2 \ /t\
1 + tan (1) \1 + tan (1)/*tan|-|
\2/
$$\frac{\left(- 2 t + 2 \pi\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}} + \frac{\left(- t + \frac{\pi}{2}\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right) \tan{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right) \tan{\left(\frac{t}{2} \right)}}$$
/ / 2/t\\ / 2/t\\\
| t*|1 + tan |-|| pi*|1 + tan |-|||
| \ \2// \ \2//|
2*|pi - t - --------------- + ----------------|*tan(1)
| /t\ /t\ |
| 2*tan|-| 4*tan|-| |
\ \2/ \2/ /
------------------------------------------------------
2
1 + tan (1)
$$\frac{2 \left(- \frac{t \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)}{2 \tan{\left(\frac{t}{2} \right)}} - t + \frac{\pi \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)}{4 \tan{\left(\frac{t}{2} \right)}} + \pi\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}}$$
// zoo for t mod pi = 0\
|| |
(pi - 2*t)*|< 1 |*tan(1)
||------ otherwise |
(-2*t + 2*pi)*sin(2) \\sin(t) /
-------------------- + ---------------------------------------------
2 2
1 + tan (1)
$$\left(\frac{\left(- 2 t + \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin{\left(t \right)}} & \text{otherwise} \end{cases}\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}}\right) + \frac{\left(- 2 t + 2 \pi\right) \sin{\left(2 \right)}}{2}$$
// zoo for t mod pi = 0\
|| |
(pi - 2*t)*|< / pi\ |
||sec|t - --| otherwise |
-2*t + 2*pi \\ \ 2 / /
------------- + -------------------------------------------
/ pi\ / pi\
2*sec|2 - --| 2*sec|2 - --|
\ 2 / \ 2 /
$$\left(\frac{\left(- 2 t + \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\sec{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{2 \sec{\left(- \frac{\pi}{2} + 2 \right)}}\right) + \frac{- 2 t + 2 \pi}{2 \sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
/pi \ / pi\
2*|-- - t|*sec(1)*sec|t - --|
-2*t + 2*pi \2 / \ 2 /
------------- + ------------------------------
/ pi\ / 2 \
2*sec|2 - --| | sec (1) | / pi\
\ 2 / |1 + ------------|*sec|1 - --|
| 2/ pi\| \ 2 /
| sec |1 - --||
\ \ 2 //
$$\frac{2 \left(- t + \frac{\pi}{2}\right) \sec{\left(1 \right)} \sec{\left(t - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(1 \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + 1 \right)}}\right) \sec{\left(- \frac{\pi}{2} + 1 \right)}} + \frac{- 2 t + 2 \pi}{2 \sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
// zoo for t mod pi = 0\
|| |
|| /t\ |
(pi - 2*t)*|< -tan|-| |*sin(2)
|| \2/ |
||----------- otherwise |
(-2*t + 2*pi)*sin(2) \\-1 + cos(t) /
-------------------- + --------------------------------------------------
2 2
$$\left(\frac{\left(- 2 t + \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\- \frac{\tan{\left(\frac{t}{2} \right)}}{\cos{\left(t \right)} - 1} & \text{otherwise} \end{cases}\right) \sin{\left(2 \right)}}{2}\right) + \frac{\left(- 2 t + 2 \pi\right) \sin{\left(2 \right)}}{2}$$
/ pi\ /pi \ / pi\
(-2*t + 2*pi)*cos|2 - --| 2*|-- - t|*cos|1 - --|
\ 2 / \2 / \ 2 /
------------------------- + -------------------------------------
2 / 2/ pi\\
| cos |1 - --||
| \ 2 /| / pi\
|1 + ------------|*cos(1)*cos|t - --|
| 2 | \ 2 /
\ cos (1) /
$$\frac{\left(- 2 t + 2 \pi\right) \cos{\left(- \frac{\pi}{2} + 2 \right)}}{2} + \frac{2 \left(- t + \frac{\pi}{2}\right) \cos{\left(- \frac{\pi}{2} + 1 \right)}}{\left(1 + \frac{\cos^{2}{\left(- \frac{\pi}{2} + 1 \right)}}{\cos^{2}{\left(1 \right)}}\right) \cos{\left(1 \right)} \cos{\left(t - \frac{\pi}{2} \right)}}$$
/ 1 \ /pi \ /t\
|1 + -------|*|-- - t|*cot|-|
| 2/t\| \2 / \2/
| cot |-||
2*(pi - t) \ \2//
-------------------- + -----------------------------
/ 1 \ / 1 \
|1 + -------|*cot(1) |1 + -------|*cot(1)
| 2 | | 2 |
\ cot (1)/ \ cot (1)/
$$\frac{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right) \left(- t + \frac{\pi}{2}\right) \cot{\left(\frac{t}{2} \right)}}{\left(1 + \frac{1}{\cot^{2}{\left(1 \right)}}\right) \cot{\left(1 \right)}} + \frac{2 \left(- t + \pi\right)}{\left(1 + \frac{1}{\cot^{2}{\left(1 \right)}}\right) \cot{\left(1 \right)}}$$
// zoo for t mod pi = 0\
|| |
|| 1 | / pi\
(pi - 2*t)*|<----------- otherwise |*cos|2 - --|
/ pi\ || / pi\ | \ 2 /
(-2*t + 2*pi)*cos|2 - --| ||cos|t - --| |
\ 2 / \\ \ 2 / /
------------------------- + -------------------------------------------------------
2 2
$$\left(\frac{\left(- 2 t + \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\cos{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \cos{\left(- \frac{\pi}{2} + 2 \right)}}{2}\right) + \frac{\left(- 2 t + 2 \pi\right) \cos{\left(- \frac{\pi}{2} + 2 \right)}}{2}$$
// zoo for t mod pi = 0\
|| |
||/ zoo for t mod pi = 0 |
(pi - 2*t)*|<| |*sin(2)
||< 1 otherwise |
|||------ otherwise |
(-2*t + 2*pi)*sin(2) \\\sin(t) /
-------------------- + ----------------------------------------------------------------
2 2
$$\left(\frac{\left(- 2 t + \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin{\left(t \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \sin{\left(2 \right)}}{2}\right) + \frac{\left(- 2 t + 2 \pi\right) \sin{\left(2 \right)}}{2}$$
// / 3*pi\ \
|| 1 for |t + ----| mod 2*pi = 0|
|| \ 2 / |
|| |
/pi \ || /t\ |
2*|-- - t|*|< tan|-| |*tan(1)
\2 / || 1 \2/ |
||-------- + ------ otherwise |
|| /t\ 2 |
||2*tan|-| |
\\ \2/ /
(pi - t)*sin(2) + -------------------------------------------------------------------
2
1 + tan (1)
$$\left(\frac{2 \left(- t + \frac{\pi}{2}\right) \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan{\left(\frac{t}{2} \right)}}{2} + \frac{1}{2 \tan{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}}\right) + \left(- t + \pi\right) \sin{\left(2 \right)}$$
/ // zoo for t mod pi = 0\ // zoo for t mod pi = 0\\
| || | || ||
|-2*t + 2*pi + pi*|< 1 | - 2*t*|< 1 ||*sin(2)
| ||------ otherwise | ||------ otherwise ||
\ \\sin(t) / \\sin(t) //
---------------------------------------------------------------------------------------
2
$$\frac{\left(\left(- 2 t \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin{\left(t \right)}} & \text{otherwise} \end{cases}\right)\right) - 2 t + \left(\pi \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin{\left(t \right)}} & \text{otherwise} \end{cases}\right)\right) + 2 \pi\right) \sin{\left(2 \right)}}{2}$$
// zoo for t mod pi = 0\
|| |
|| 2/t\ |
||1 + tan |-| |
(pi - 2*t)*|< \2/ |*tan(1)
||----------- otherwise |
|| /t\ |
|| 2*tan|-| |
(-2*t + 2*pi)*tan(1) \\ \2/ /
-------------------- + --------------------------------------------------
2 2
1 + tan (1) 1 + tan (1)
$$\left(\frac{\left(- 2 t + \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\tan^{2}{\left(\frac{t}{2} \right)} + 1}{2 \tan{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}}\right) + \frac{\left(- 2 t + 2 \pi\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}}$$
// zoo for t mod pi = 0\
|| |
|| 2/t\ |
/pi \ ||1 + cot |-| |
2*|-- - t|*|< \2/ |*cot(1)
\2 / ||----------- otherwise |
|| /t\ |
|| 2*cot|-| |
2*(pi - t)*cot(1) \\ \2/ /
----------------- + --------------------------------------------------
2 2
1 + cot (1) 1 + cot (1)
$$\left(\frac{2 \left(- t + \frac{\pi}{2}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} + 1}{2 \cot{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \cot{\left(1 \right)}}{\cot^{2}{\left(1 \right)} + 1}\right) + \frac{2 \left(- t + \pi\right) \cot{\left(1 \right)}}{\cot^{2}{\left(1 \right)} + 1}$$
// zoo for t mod pi = 0\
|| |
|| 2/t\ |
/pi \ ||1 + cot |-| |
2*|-- - t|*|< \2/ |
\2 / ||----------- otherwise |
|| /t\ |
|| 2*cot|-| |
(-2*t + 2*pi)*cot(1) \\ \2/ /
-------------------- + -------------------------------------------
2 / 1 \
1 + cot (1) |1 + -------|*cot(1)
| 2 |
\ cot (1)/
$$\left(\frac{2 \left(- t + \frac{\pi}{2}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} + 1}{2 \cot{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(1 \right)}}\right) \cot{\left(1 \right)}}\right) + \frac{\left(- 2 t + 2 \pi\right) \cot{\left(1 \right)}}{\cot^{2}{\left(1 \right)} + 1}$$
/ 2/ pi\\ / t pi\
|1 - cot |1 + --||*(1 + sin(2))*|- - + --|
/ 2/ pi\\ /pi t\ \ \ 4 // \ 2 4 /
|1 - cot |1 + --||*(1 + sin(2))*|-- - -| + ------------------------------------------
\ \ 4 // \2 2/ / 2/t pi\\ 2/t pi\
|1 - cot |- + --||*sin |- + --|
\ \2 4 // \2 4 /
$$\left(- \cot^{2}{\left(\frac{\pi}{4} + 1 \right)} + 1\right) \left(- \frac{t}{2} + \frac{\pi}{2}\right) \left(\sin{\left(2 \right)} + 1\right) + \frac{\left(- \cot^{2}{\left(\frac{\pi}{4} + 1 \right)} + 1\right) \left(- \frac{t}{2} + \frac{\pi}{4}\right) \left(\sin{\left(2 \right)} + 1\right)}{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}$$
/ 2/ pi\\ / 2/t pi\\ / 2/ pi\\ /pi \
|-1 + tan |1 + --||*(pi - t) |1 + tan |- + --||*|-1 + tan |1 + --||*|-- - t|
\ \ 4 // \ \2 4 // \ \ 4 // \2 /
---------------------------- + -----------------------------------------------
2/ pi\ / 2/ pi\\ / 2/t pi\\
1 + tan |1 + --| |1 + tan |1 + --||*|-1 + tan |- + --||
\ 4 / \ \ 4 // \ \2 4 //
$$\frac{\left(-1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(- t + \pi\right)}{1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}} + \frac{\left(-1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(- t + \frac{\pi}{2}\right) \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)}$$
/ 2/ pi\\ / 2/t pi\\ / 2/ pi\\ /pi \
|1 - cot |1 + --||*(pi - t) |1 + cot |- + --||*|1 - cot |1 + --||*|-- - t|
\ \ 4 // \ \2 4 // \ \ 4 // \2 /
--------------------------- + ----------------------------------------------
2/ pi\ / 2/ pi\\ / 2/t pi\\
1 + cot |1 + --| |1 + cot |1 + --||*|1 - cot |- + --||
\ 4 / \ \ 4 // \ \2 4 //
$$\frac{\left(- \cot^{2}{\left(\frac{\pi}{4} + 1 \right)} + 1\right) \left(- t + \pi\right)}{\cot^{2}{\left(\frac{\pi}{4} + 1 \right)} + 1} + \frac{\left(- \cot^{2}{\left(\frac{\pi}{4} + 1 \right)} + 1\right) \left(- t + \frac{\pi}{2}\right) \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{\pi}{4} + 1 \right)} + 1\right)}$$
// zoo for t mod pi = 0\
|| |
||/ 1 \ /t\ |
/pi \ |||1 + -------|*tan|-| |
2*|-- - t|*|<| 2/t\| \2/ |
\2 / ||| tan |-|| |
||\ \2// |
||-------------------- otherwise |
2*(pi - t) \\ 2 /
-------------------- + ----------------------------------------------------
/ 1 \ / 1 \
|1 + -------|*tan(1) |1 + -------|*tan(1)
| 2 | | 2 |
\ tan (1)/ \ tan (1)/
$$\left(\frac{2 \left(- t + \frac{\pi}{2}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right) \tan{\left(\frac{t}{2} \right)}}{2} & \text{otherwise} \end{cases}\right)}{\left(\frac{1}{\tan^{2}{\left(1 \right)}} + 1\right) \tan{\left(1 \right)}}\right) + \frac{2 \left(- t + \pi\right)}{\left(\frac{1}{\tan^{2}{\left(1 \right)}} + 1\right) \tan{\left(1 \right)}}$$
/ 4/t\\
| 4*sin |-||
2 | \2/|
sin (1)*|1 + ---------|*(pi - 2*t)*sin(t)
| 2 |
(-4*t + 4*pi)*sin(2) \ sin (t) /
-------------------- + -----------------------------------------
2 / 4 \
2 sin (2) | 4*sin (1)|
4*sin (1) + ------- (1 - cos(t))*|1 + ---------|*sin(2)
2 | 2 |
sin (1) \ sin (2) /
$$\frac{\left(- 2 t + \pi\right) \left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right) \sin^{2}{\left(1 \right)} \sin{\left(t \right)}}{\left(1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}\right) \left(- \cos{\left(t \right)} + 1\right) \sin{\left(2 \right)}} + \frac{\left(- 4 t + 4 \pi\right) \sin{\left(2 \right)}}{\frac{\sin^{2}{\left(2 \right)}}{\sin^{2}{\left(1 \right)}} + 4 \sin^{2}{\left(1 \right)}}$$
// zoo for t mod pi = 0\
|| |
||/ zoo for t mod pi = 0 |
||| |
||| 2/t\ |
(pi - 2*t)*|<|1 + cot |-| |*cot(1)
||< \2/ otherwise |
|||----------- otherwise |
||| /t\ |
||| 2*cot|-| |
(-2*t + 2*pi)*cot(1) \\\ \2/ /
-------------------- + ---------------------------------------------------------------------
2 2
1 + cot (1) 1 + cot (1)
$$\left(\frac{\left(- 2 t + \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} + 1}{2 \cot{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \cot{\left(1 \right)}}{\cot^{2}{\left(1 \right)} + 1}\right) + \frac{\left(- 2 t + 2 \pi\right) \cot{\left(1 \right)}}{\cot^{2}{\left(1 \right)} + 1}$$
/ 4/t\\
| 4*sin |-||
2 | \2/| /pi \
sin (1)*|1 + ---------|*|-- - t|*sin(t)
2 | 2 | \2 /
4*sin (1)*(pi - t) \ sin (t) /
---------------------- + ---------------------------------------
/ 4 \ / 4 \
| 4*sin (1)| | 4*sin (1)| 2/t\
|1 + ---------|*sin(2) |1 + ---------|*sin(2)*sin |-|
| 2 | | 2 | \2/
\ sin (2) / \ sin (2) /
$$\frac{4 \left(- t + \pi\right) \sin^{2}{\left(1 \right)}}{\left(1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}\right) \sin{\left(2 \right)}} + \frac{\left(- t + \frac{\pi}{2}\right) \left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right) \sin^{2}{\left(1 \right)} \sin{\left(t \right)}}{\left(1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}\right) \sin{\left(2 \right)} \sin^{2}{\left(\frac{t}{2} \right)}}$$
/ // zoo for t mod pi = 0\ \
| || | |
| || 2/t\ | |
| ||1 + cot |-| | |
| pi*|< \2/ | |
| ||----------- otherwise | // zoo for t mod pi = 0\|
| || /t\ | || ||
| || 2*cot|-| | || 2/t\ ||
| \\ \2/ / ||1 + cot |-| ||
2*|pi - t + ----------------------------------- - t*|< \2/ ||*cot(1)
| 2 ||----------- otherwise ||
| || /t\ ||
| || 2*cot|-| ||
\ \\ \2/ //
--------------------------------------------------------------------------------------------
2
1 + cot (1)
$$\frac{2 \left(\left(- t \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} + 1}{2 \cot{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - t + \left(\frac{\pi \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} + 1}{2 \cot{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)}{2}\right) + \pi\right) \cot{\left(1 \right)}}{\cot^{2}{\left(1 \right)} + 1}$$
// / 3*pi\ \
|| 1 for |t + ----| mod 2*pi = 0|
|| \ 2 / |
|| |
/ 2/ pi\\ /pi \ || 2/t pi\ |
|-1 + tan |1 + --||*|-- - t|*|< 1 + tan |- + --| |
\ \ 4 // \2 / || \2 4 / |
||----------------- otherwise |
/ 2/ pi\\ || 2/t pi\ |
|-1 + tan |1 + --||*(pi - t) ||-1 + tan |- + --| |
\ \ 4 // \\ \2 4 / /
---------------------------- + ------------------------------------------------------------------------------
2/ pi\ 2/ pi\
1 + tan |1 + --| 1 + tan |1 + --|
\ 4 / \ 4 /
$$\left(\frac{\left(-1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(- t + \frac{\pi}{2}\right) \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1} & \text{otherwise} \end{cases}\right)}{1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}}\right) + \frac{\left(-1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(- t + \pi\right)}{1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}}$$
// zoo for t mod pi = 0\
|| |
|| 2/t\ |
|| sin |-| |
(-4*t + 2*pi)*|< 1 \2/ |*sin(2)
||-------- + ------- otherwise |
|| /t\ sin(t) |
||2*tan|-| |
(-4*t + 4*pi)*sin(2) \\ \2/ /
--------------------- + ------------------------------------------------------------
/ 2 \ / 2 \
| sin (2)| 2 | sin (2)| 2
|4 + -------|*sin (1) |4 + -------|*sin (1)
| 4 | | 4 |
\ sin (1)/ \ sin (1)/
$$\left(\frac{\left(- 4 t + 2 \pi\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(\frac{t}{2} \right)}}{\sin{\left(t \right)}} + \frac{1}{2 \tan{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \sin{\left(2 \right)}}{\left(\frac{\sin^{2}{\left(2 \right)}}{\sin^{4}{\left(1 \right)}} + 4\right) \sin^{2}{\left(1 \right)}}\right) + \frac{\left(- 4 t + 4 \pi\right) \sin{\left(2 \right)}}{\left(\frac{\sin^{2}{\left(2 \right)}}{\sin^{4}{\left(1 \right)}} + 4\right) \sin^{2}{\left(1 \right)}}$$
/ 2/t\ \
| sec |-| |
| \2/ | /pi \ /t pi\
|1 + ------------|*|-- - t|*sec(1)*sec|- - --|
| 2/t pi\| \2 / \2 2 /
| sec |- - --||
2*(pi - t)*sec(1) \ \2 2 //
------------------------------ + ----------------------------------------------
/ 2 \ / 2 \
| sec (1) | / pi\ | sec (1) | /t\ / pi\
|1 + ------------|*sec|1 - --| |1 + ------------|*sec|-|*sec|1 - --|
| 2/ pi\| \ 2 / | 2/ pi\| \2/ \ 2 /
| sec |1 - --|| | sec |1 - --||
\ \ 2 // \ \ 2 //
$$\frac{\left(- t + \frac{\pi}{2}\right) \left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(1 \right)} \sec{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(1 \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + 1 \right)}}\right) \sec{\left(\frac{t}{2} \right)} \sec{\left(- \frac{\pi}{2} + 1 \right)}} + \frac{2 \left(- t + \pi\right) \sec{\left(1 \right)}}{\left(1 + \frac{\sec^{2}{\left(1 \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + 1 \right)}}\right) \sec{\left(- \frac{\pi}{2} + 1 \right)}}$$
/ 2/t pi\\
| cos |- - --||
| \2 2 /| /pi \ /t\ / pi\
|1 + ------------|*|-- - t|*cos|-|*cos|1 - --|
/ pi\ | 2/t\ | \2 / \2/ \ 2 /
2*(pi - t)*cos|1 - --| | cos |-| |
\ 2 / \ \2/ /
------------------------- + ----------------------------------------------
/ 2/ pi\\ / 2/ pi\\
| cos |1 - --|| | cos |1 - --||
| \ 2 /| | \ 2 /| /t pi\
|1 + ------------|*cos(1) |1 + ------------|*cos(1)*cos|- - --|
| 2 | | 2 | \2 2 /
\ cos (1) / \ cos (1) /
$$\frac{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right) \left(- t + \frac{\pi}{2}\right) \cos{\left(\frac{t}{2} \right)} \cos{\left(- \frac{\pi}{2} + 1 \right)}}{\left(1 + \frac{\cos^{2}{\left(- \frac{\pi}{2} + 1 \right)}}{\cos^{2}{\left(1 \right)}}\right) \cos{\left(1 \right)} \cos{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + \frac{2 \left(- t + \pi\right) \cos{\left(- \frac{\pi}{2} + 1 \right)}}{\left(1 + \frac{\cos^{2}{\left(- \frac{\pi}{2} + 1 \right)}}{\cos^{2}{\left(1 \right)}}\right) \cos{\left(1 \right)}}$$
/ 2/pi t\\
| csc |-- - -||
| \2 2/| /pi \ /t\ / pi\
|1 + ------------|*|-- - t|*csc|-|*csc|-1 + --|
/ pi\ | 2/t\ | \2 / \2/ \ 2 /
2*(pi - t)*csc|-1 + --| | csc |-| |
\ 2 / \ \2/ /
-------------------------- + -----------------------------------------------
/ 2/ pi\\ / 2/ pi\\
| csc |-1 + --|| | csc |-1 + --||
| \ 2 /| | \ 2 /| /pi t\
|1 + -------------|*csc(1) |1 + -------------|*csc(1)*csc|-- - -|
| 2 | | 2 | \2 2/
\ csc (1) / \ csc (1) /
$$\frac{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right) \left(- t + \frac{\pi}{2}\right) \csc{\left(\frac{t}{2} \right)} \csc{\left(-1 + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(1 \right)}}\right) \csc{\left(1 \right)} \csc{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + \frac{2 \left(- t + \pi\right) \csc{\left(-1 + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(1 \right)}}\right) \csc{\left(1 \right)}}$$
// zoo for t mod pi = 0\
|| |
|| / 2 \ |
|| 2/t\ | sin (t) | |
/pi \ ||sin |-|*|1 + ---------| |
|-- - t|*|< \2/ | 4/t\| |*sin(2)
\2 / || | 4*sin |-|| |
|| \ \2// |
||----------------------- otherwise |
|| sin(t) |
(pi - t)*sin(2) \\ /
----------------------- + ------------------------------------------------------------
/ 2 \ / 2 \
| sin (2) | 2 | sin (2) | 2
|1 + ---------|*sin (1) |1 + ---------|*sin (1)
| 4 | | 4 |
\ 4*sin (1)/ \ 4*sin (1)/
$$\left(\frac{\left(- t + \frac{\pi}{2}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right) \sin^{2}{\left(\frac{t}{2} \right)}}{\sin{\left(t \right)}} & \text{otherwise} \end{cases}\right) \sin{\left(2 \right)}}{\left(\frac{\sin^{2}{\left(2 \right)}}{4 \sin^{4}{\left(1 \right)}} + 1\right) \sin^{2}{\left(1 \right)}}\right) + \frac{\left(- t + \pi\right) \sin{\left(2 \right)}}{\left(\frac{\sin^{2}{\left(2 \right)}}{4 \sin^{4}{\left(1 \right)}} + 1\right) \sin^{2}{\left(1 \right)}}$$
// zoo for t mod pi = 0\
|| |
||/ 2/t pi\\ |
||| sec |- - --|| |
||| \2 2 /| /t\ |
/pi \ |||1 + ------------|*sec|-| | / pi\
2*|-- - t|*|<| 2/t\ | \2/ |*sec|1 - --|
\2 / ||| sec |-| | | \ 2 /
||\ \2/ / |
||------------------------- otherwise |
/ pi\ || /t pi\ |
2*(pi - t)*sec|1 - --| || 2*sec|- - --| |
\ 2 / \\ \2 2 / /
------------------------- + ---------------------------------------------------------------------
/ 2/ pi\\ / 2/ pi\\
| sec |1 - --|| | sec |1 - --||
| \ 2 /| | \ 2 /|
|1 + ------------|*sec(1) |1 + ------------|*sec(1)
| 2 | | 2 |
\ sec (1) / \ sec (1) /
$$\left(\frac{2 \left(- t + \frac{\pi}{2}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right) \sec{\left(\frac{t}{2} \right)}}{2 \sec{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \sec{\left(- \frac{\pi}{2} + 1 \right)}}{\left(\frac{\sec^{2}{\left(- \frac{\pi}{2} + 1 \right)}}{\sec^{2}{\left(1 \right)}} + 1\right) \sec{\left(1 \right)}}\right) + \frac{2 \left(- t + \pi\right) \sec{\left(- \frac{\pi}{2} + 1 \right)}}{\left(\frac{\sec^{2}{\left(- \frac{\pi}{2} + 1 \right)}}{\sec^{2}{\left(1 \right)}} + 1\right) \sec{\left(1 \right)}}$$
// zoo for t mod pi = 0\
|| |
||/ 2/t\ \ |
||| csc |-| | |
||| \2/ | /pi t\ |
/pi \ |||1 + ------------|*csc|-- - -| |
2*|-- - t|*|<| 2/pi t\| \2 2/ |*csc(1)
\2 / ||| csc |-- - -|| |
||\ \2 2// |
||------------------------------ otherwise |
|| /t\ |
|| 2*csc|-| |
2*(pi - t)*csc(1) \\ \2/ /
-------------------------------- + ---------------------------------------------------------------------
/ 2 \ / 2 \
| csc (1) | / pi\ | csc (1) | / pi\
|1 + -------------|*csc|-1 + --| |1 + -------------|*csc|-1 + --|
| 2/ pi\| \ 2 / | 2/ pi\| \ 2 /
| csc |-1 + --|| | csc |-1 + --||
\ \ 2 // \ \ 2 //
$$\left(\frac{2 \left(- t + \frac{\pi}{2}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \csc{\left(1 \right)}}{\left(\frac{\csc^{2}{\left(1 \right)}}{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(-1 + \frac{\pi}{2} \right)}}\right) + \frac{2 \left(- t + \pi\right) \csc{\left(1 \right)}}{\left(\frac{\csc^{2}{\left(1 \right)}}{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(-1 + \frac{\pi}{2} \right)}}$$
// zoo for t mod pi = 0\
|| |
||/ 2/t\ \ |
||| cos |-| | |
||| \2/ | /t pi\ |
/pi \ |||1 + ------------|*cos|- - --| |
2*|-- - t|*|<| 2/t pi\| \2 2 / |*cos(1)
\2 / ||| cos |- - --|| |
||\ \2 2 // |
||------------------------------ otherwise |
|| /t\ |
|| 2*cos|-| |
2*(pi - t)*cos(1) \\ \2/ /
------------------------------ + ---------------------------------------------------------------------
/ 2 \ / 2 \
| cos (1) | / pi\ | cos (1) | / pi\
|1 + ------------|*cos|1 - --| |1 + ------------|*cos|1 - --|
| 2/ pi\| \ 2 / | 2/ pi\| \ 2 /
| cos |1 - --|| | cos |1 - --||
\ \ 2 // \ \ 2 //
$$\left(\frac{2 \left(- t + \frac{\pi}{2}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \cos{\left(1 \right)}}{\left(\frac{\cos^{2}{\left(1 \right)}}{\cos^{2}{\left(- \frac{\pi}{2} + 1 \right)}} + 1\right) \cos{\left(- \frac{\pi}{2} + 1 \right)}}\right) + \frac{2 \left(- t + \pi\right) \cos{\left(1 \right)}}{\left(\frac{\cos^{2}{\left(1 \right)}}{\cos^{2}{\left(- \frac{\pi}{2} + 1 \right)}} + 1\right) \cos{\left(- \frac{\pi}{2} + 1 \right)}}$$
2*(pi - t)*cos(1)/((1 + cos(1)^2/cos(1 - pi/2)^2)*cos(1 - pi/2)) + 2*(pi/2 - t)*Piecewise((±oo, Mod(t = pi, 0)), ((1 + cos(t/2)^2/cos(t/2 - pi/2)^2)*cos(t/2 - pi/2)/(2*cos(t/2)), True))*cos(1)/((1 + cos(1)^2/cos(1 - pi/2)^2)*cos(1 - pi/2))
Численный ответ
[src]
2.85664211604366 - 0.909297426825682*t + 0.909297426825682*(1.5707963267949 - t)/sin(pi - t)
2.85664211604366 - 0.909297426825682*t + 0.909297426825682*(1.5707963267949 - t)/sin(pi - t)