Господин Экзамен
cos(pi+3*t) если t=1
Решение
Подстановка условия
[src]
cos(pi + 3*t) при t = 1
подставляем
cos(pi + 3*t)
$$\cos{\left(3 t + \pi \right)}$$
-cos(3*t)
$$- \cos{\left(3 t \right)}$$
переменные
t = 1
$$t = 1$$
-cos(3*(1))
$$- \cos{\left(3 (1) \right)}$$
-cos(3*1)
$$- \cos{\left(3 \cdot 1 \right)}$$
-cos(3)
$$- \cos{\left(3 \right)}$$
-cos(3)
Степени
[src]
-cos(3*t)
$$- \cos{\left(3 t \right)}$$
I*(pi + 3*t) I*(-pi - 3*t)
e e
------------- + --------------
2 2
$$\frac{e^{i \left(- 3 t - \pi\right)}}{2} + \frac{e^{i \left(3 t + \pi\right)}}{2}$$
exp(i*(pi + 3*t))/2 + exp(i*(-pi - 3*t))/2
Тригонометрическая часть
[src]
-cos(3*t)
$$- \cos{\left(3 t \right)}$$
-1 -------- sec(3*t)
$$- \frac{1}{\sec{\left(3 t \right)}}$$
/pi \
-sin|-- + 3*t|
\2 /
$$- \sin{\left(3 t + \frac{\pi}{2} \right)}$$
-1 ------------- /pi \ csc|-- - 3*t| \2 /
$$- \frac{1}{\csc{\left(- 3 t + \frac{\pi}{2} \right)}}$$
/ 2/3*t\\
-|-1 + cot |---||
\ \ 2 //
------------------
2/3*t\
1 + cot |---|
\ 2 /
$$- \frac{\cot^{2}{\left(\frac{3 t}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 t}{2} \right)} + 1}$$
/ 2/3*t\\
-|1 - tan |---||
\ \ 2 //
-----------------
2/3*t\
1 + tan |---|
\ 2 /
$$- \frac{- \tan^{2}{\left(\frac{3 t}{2} \right)} + 1}{\tan^{2}{\left(\frac{3 t}{2} \right)} + 1}$$
// 1 for 3*t mod 2*pi = 0\ -|< | \\cos(3*t) otherwise /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\cos{\left(3 t \right)} & \text{otherwise} \end{cases}$$
/ 1 \
-|1 - ---------|
| 2/3*t\|
| cot |---||
\ \ 2 //
-----------------
1
1 + ---------
2/3*t\
cot |---|
\ 2 /
$$- \frac{1 - \frac{1}{\cot^{2}{\left(\frac{3 t}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{3 t}{2} \right)}}}$$
/pi 3*t\
-2*tan|-- + ---|
\4 2 /
------------------
2/pi 3*t\
1 + tan |-- + ---|
\4 2 /
$$- \frac{2 \tan{\left(\frac{3 t}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{3 t}{2} + \frac{\pi}{4} \right)} + 1}$$
// 1 for 3*t mod 2*pi = 0\ || | -|< 1 | ||-------- otherwise | \\sec(3*t) /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(3 t \right)}} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | -|< /pi \ | ||sin|-- + 3*t| otherwise | \\ \2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\sin{\left(3 t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | || 1 | -|<------------- otherwise | || /pi \ | ||csc|-- - 3*t| | \\ \2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 3 t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$
-2*(-1 - cos(6*t) + 2*cos(3*t))
--------------------------------
2
1 - cos(6*t) + 2*(1 - cos(3*t))
$$- \frac{2 \cdot \left(2 \cos{\left(3 t \right)} - \cos{\left(6 t \right)} - 1\right)}{2 \left(- \cos{\left(3 t \right)} + 1\right)^{2} - \cos{\left(6 t \right)} + 1}$$
/ 4/3*t\\
| 4*sin |---||
| \ 2 /|
-|1 - -----------|
| 2 |
\ sin (3*t) /
-------------------
4/3*t\
4*sin |---|
\ 2 /
1 + -----------
2
sin (3*t)
$$- \frac{- \frac{4 \sin^{4}{\left(\frac{3 t}{2} \right)}}{\sin^{2}{\left(3 t \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{3 t}{2} \right)}}{\sin^{2}{\left(3 t \right)}} + 1}$$
// 1 for 3*t mod 2*pi = 0\ || | || 2/3*t\ | ||-1 + cot |---| | -|< \ 2 / | ||-------------- otherwise | || 2/3*t\ | ||1 + cot |---| | \\ \ 2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 t}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 t}{2} \right)} + 1} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | || 2/3*t\ | ||1 - tan |---| | -|< \ 2 / | ||------------- otherwise | || 2/3*t\ | ||1 + tan |---| | \\ \ 2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{3 t}{2} \right)} + 1}{\tan^{2}{\left(\frac{3 t}{2} \right)} + 1} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\
|| |
-|
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\cos{\left(3 t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | || 1 | ||-1 + --------- | || 2/3*t\ | || tan |---| | -|< \ 2 / | ||-------------- otherwise | || 1 | ||1 + --------- | || 2/3*t\ | || tan |---| | \\ \ 2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{3 t}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{3 t}{2} \right)}}} & \text{otherwise} \end{cases}$$
// /pi \ \ || 0 for |-- + 3*t| mod pi = 0| || \2 / | -|< | || /pi 3*t\ | ||(1 + sin(3*t))*cot|-- + ---| otherwise | \\ \4 2 / /
$$- \begin{cases} 0 & \text{for}\: \left(3 t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(3 t \right)} + 1\right) \cot{\left(\frac{3 t}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}$$
/ 2/ pi 3*t\\
| cos |- -- + ---||
| \ 2 2 /|
-|1 - ----------------|
| 2/3*t\ |
| cos |---| |
\ \ 2 / /
------------------------
2/ pi 3*t\
cos |- -- + ---|
\ 2 2 /
1 + ----------------
2/3*t\
cos |---|
\ 2 /
$$- \frac{1 - \frac{\cos^{2}{\left(\frac{3 t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 t}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{3 t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 t}{2} \right)}}}$$
/ 2/3*t\ \
| sec |---| |
| \ 2 / |
-|1 - ----------------|
| 2/ pi 3*t\|
| sec |- -- + ---||
\ \ 2 2 //
------------------------
2/3*t\
sec |---|
\ 2 /
1 + ----------------
2/ pi 3*t\
sec |- -- + ---|
\ 2 2 /
$$- \frac{- \frac{\sec^{2}{\left(\frac{3 t}{2} \right)}}{\sec^{2}{\left(\frac{3 t}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{3 t}{2} \right)}}{\sec^{2}{\left(\frac{3 t}{2} - \frac{\pi}{2} \right)}} + 1}$$
/ 2/pi 3*t\\
| csc |-- - ---||
| \2 2 /|
-|1 - --------------|
| 2/3*t\ |
| csc |---| |
\ \ 2 / /
----------------------
2/pi 3*t\
csc |-- - ---|
\2 2 /
1 + --------------
2/3*t\
csc |---|
\ 2 /
$$- \frac{1 - \frac{\csc^{2}{\left(- \frac{3 t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 t}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{3 t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 t}{2} \right)}}}$$
// /pi \ \ || 0 for |-- + 3*t| mod pi = 0| || \2 / | || | || /pi 3*t\ | -|< 2*cot|-- + ---| | || \4 2 / | ||------------------ otherwise | || 2/pi 3*t\ | ||1 + cot |-- + ---| | \\ \4 2 / /
$$- \begin{cases} 0 & \text{for}\: \left(3 t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 t}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{3 t}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | || -2 - 2*cos(6*t) + 4*cos(3*t) | -|<-------------------------------- otherwise | || 2 | ||1 - cos(6*t) + 2*(1 - cos(3*t)) | \\ /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{4 \cos{\left(3 t \right)} - 2 \cos{\left(6 t \right)} - 2}{2 \left(- \cos{\left(3 t \right)} + 1\right)^{2} - \cos{\left(6 t \right)} + 1} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | || 2 | || sin (3*t) | ||-1 + ----------- | || 4/3*t\ | || 4*sin |---| | -|< \ 2 / | ||---------------- otherwise | || 2 | || sin (3*t) | ||1 + ----------- | || 4/3*t\ | || 4*sin |---| | \\ \ 2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(3 t \right)}}{4 \sin^{4}{\left(\frac{3 t}{2} \right)}}}{1 + \frac{\sin^{2}{\left(3 t \right)}}{4 \sin^{4}{\left(\frac{3 t}{2} \right)}}} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | ||/ 1 for 3*t mod 2*pi = 0 | ||| | ||| 2/3*t\ | -|<|-1 + cot |---| | ||< \ 2 / otherwise | |||-------------- otherwise | ||| 2/3*t\ | |||1 + cot |---| | \\\ \ 2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 t}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 t}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | || 2/3*t\ | || cos |---| | || \ 2 / | ||-1 + ---------------- | || 2/ pi 3*t\ | || cos |- -- + ---| | -|< \ 2 2 / | ||--------------------- otherwise | || 2/3*t\ | || cos |---| | || \ 2 / | || 1 + ---------------- | || 2/ pi 3*t\ | || cos |- -- + ---| | \\ \ 2 2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{3 t}{2} \right)}}{\cos^{2}{\left(\frac{3 t}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{3 t}{2} \right)}}{\cos^{2}{\left(\frac{3 t}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | || 2/ pi 3*t\ | || sec |- -- + ---| | || \ 2 2 / | ||-1 + ---------------- | || 2/3*t\ | || sec |---| | -|< \ 2 / | ||--------------------- otherwise | || 2/ pi 3*t\ | || sec |- -- + ---| | || \ 2 2 / | || 1 + ---------------- | || 2/3*t\ | || sec |---| | \\ \ 2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{3 t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 t}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{3 t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 t}{2} \right)}}} & \text{otherwise} \end{cases}$$
// 1 for 3*t mod 2*pi = 0\ || | || 2/3*t\ | || csc |---| | || \ 2 / | ||-1 + -------------- | || 2/pi 3*t\ | || csc |-- - ---| | -|< \2 2 / | ||------------------- otherwise | || 2/3*t\ | || csc |---| | || \ 2 / | || 1 + -------------- | || 2/pi 3*t\ | || csc |-- - ---| | \\ \2 2 / /
$$- \begin{cases} 1 & \text{for}\: 3 t \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{3 t}{2} \right)}}{\csc^{2}{\left(- \frac{3 t}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{3 t}{2} \right)}}{\csc^{2}{\left(- \frac{3 t}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}$$
-Piecewise((1, Mod(3*t = 2*pi, 0)), ((-1 + csc(3*t/2)^2/csc(pi/2 - 3*t/2)^2)/(1 + csc(3*t/2)^2/csc(pi/2 - 3*t/2)^2), True))