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