Господин Экзамен
2*sin(2*a) если a=2
Решение
Подстановка условия
[src]
2*sin(2*a) при a = 2
подставляем
2*sin(2*a)
$$2 \sin{\left(2 a \right)}$$
2*sin(2*a)
$$2 \sin{\left(2 a \right)}$$
переменные
a = 2
$$a = 2$$
2*sin(2*(2))
$$2 \sin{\left(2 (2) \right)}$$
2*sin(2*2)
$$2 \sin{\left(2 \cdot 2 \right)}$$
2*sin(4)
$$2 \sin{\left(4 \right)}$$
2*sin(4)
Степени
[src]
/ -2*I*a 2*I*a\ -I*\- e + e /
$$- i \left(e^{2 i a} - e^{- 2 i a}\right)$$
-i*(-exp(-2*i*a) + exp(2*i*a))
Тригонометрическая часть
[src]
2 -------- csc(2*a)
$$\frac{2}{\csc{\left(2 a \right)}}$$
/ pi\
2*cos|2*a - --|
\ 2 /
$$2 \cos{\left(2 a - \frac{\pi}{2} \right)}$$
2 ------------- csc(pi - 2*a)
$$\frac{2}{\csc{\left(- 2 a + \pi \right)}}$$
4*cos(a)*sin(a)
$$4 \sin{\left(a \right)} \cos{\left(a \right)}$$
2 ------------- / pi\ sec|2*a - --| \ 2 /
$$\frac{2}{\sec{\left(2 a - \frac{\pi}{2} \right)}}$$
2 ------------- /pi \ sec|-- - 2*a| \2 /
$$\frac{2}{\sec{\left(- 2 a + \frac{\pi}{2} \right)}}$$
4*cot(a)
-----------
2
1 + cot (a)
$$\frac{4 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}$$
4*tan(a)
-----------
2
1 + tan (a)
$$\frac{4 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}$$
4 -------------------- / 1 \ |1 + -------|*cot(a) | 2 | \ cot (a)/
$$\frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(a \right)}}\right) \cot{\left(a \right)}}$$
/ 2/ pi\\
2*|-1 + tan |a + --||
\ \ 4 //
---------------------
2/ pi\
1 + tan |a + --|
\ 4 /
$$\frac{2 \left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}$$
// 0 for 2*a mod pi = 0\ 2*|< | \\sin(2*a) otherwise /
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
/ 2/ pi\\
2*|1 - cot |a + --||
\ \ 4 //
--------------------
2/ pi\
1 + cot |a + --|
\ 4 /
$$\frac{2 \cdot \left(- \cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1}$$
// 0 for 2*a mod pi = 0\ || | 2*|< 1 | ||-------- otherwise | \\csc(2*a) /
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{1}{\csc{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ || | 2*|< / pi\ | ||cos|2*a - --| otherwise | \\ \ 2 / /
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\cos{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
2
8*sin (a)
------------------------
/ 4 \
| 4*sin (a)|
|1 + ---------|*sin(2*a)
| 2 |
\ sin (2*a)/
$$\frac{8 \sin^{2}{\left(a \right)}}{\left(\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1\right) \sin{\left(2 a \right)}}$$
// 0 for 2*a mod pi = 0\ || | || 1 | 2*|<------------- otherwise | || / pi\ | ||sec|2*a - --| | \\ \ 2 / /
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ || 1 for |2*a + ----| mod 2*pi = 0| 2*|< \ 2 / | || | \\sin(2*a) otherwise /
$$2 \left(\begin{cases} 1 & \text{for}\: \left(2 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ || | || 2*cot(a) | 2*|<----------- otherwise | || 2 | ||1 + cot (a) | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ || | || 2*tan(a) | 2*|<----------- otherwise | || 2 | ||1 + tan (a) | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)$$
4*sec(a) ------------------------------ / 2 \ | sec (a) | / pi\ |1 + ------------|*sec|a - --| | 2/ pi\| \ 2 / | sec |a - --|| \ \ 2 //
$$\frac{4 \sec{\left(a \right)}}{\left(\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
/ pi\
4*cos|a - --|
\ 2 /
-------------------------
/ 2/ pi\\
| cos |a - --||
| \ 2 /|
|1 + ------------|*cos(a)
| 2 |
\ cos (a) /
$$\frac{4 \cos{\left(a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}\right) \cos{\left(a \right)}}$$
/pi \
4*csc|-- - a|
\2 /
-------------------------
/ 2/pi \\
| csc |-- - a||
| \2 /|
|1 + ------------|*csc(a)
| 2 |
\ csc (a) /
$$\frac{4 \csc{\left(- a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right) \csc{\left(a \right)}}$$
// 0 for 2*a mod pi = 0\
|| |
|| 2 |
||-------------------- otherwise |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \tan{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\
|| |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ || 1 for |2*a + ----| mod 2*pi = 0| || \ 2 / | || | || 2/ pi\ | 2*|<-1 + tan |a + --| | || \ 4 / | ||----------------- otherwise | || 2/ pi\ | || 1 + tan |a + --| | \\ \ 4 / /
$$2 \left(\begin{cases} 1 & \text{for}\: \left(2 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\
|| |
|| sin(2*a) |
||----------------------- otherwise |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\sin{\left(2 a \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ || | ||/ 0 for 2*a mod pi = 0 | ||| | 2*|<| 2*cot(a) | ||<----------- otherwise otherwise | ||| 2 | |||1 + cot (a) | \\\ /
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\
|| |
|| / pi\ |
|| 2*sec|a - --| |
|| \ 2 / |
||------------------------- otherwise |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \sec{\left(a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ || | || 2*cos(a) | ||------------------------------ otherwise | ||/ 2 \ | 2*|<| cos (a) | / pi\ | |||1 + ------------|*cos|a - --| | ||| 2/ pi\| \ 2 / | ||| cos |a - --|| | ||\ \ 2 // | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cos{\left(a \right)}}{\left(\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ || | || 2*csc(a) | ||------------------------------ otherwise | ||/ 2 \ | 2*|<| csc (a) | /pi \ | |||1 + ------------|*csc|-- - a| | ||| 2/pi \| \2 / | ||| csc |-- - a|| | ||\ \2 // | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \csc{\left(a \right)}}{\left(\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2*Piecewise((0, Mod(2*a = pi, 0)), (2*csc(a)/((1 + csc(a)^2/csc(pi/2 - a)^2)*csc(pi/2 - a)), True))