Тригонометрическая часть
[src] // 1 for And(im(a) = 0, a mod pi = 0)\ // 0 for And(im(a) = 0, 2*a mod pi = 0)\
|| | || |
|| 2 | || 2*cot(a) |
1 - |<-1 + cot (a) | + |<----------- otherwise |
||------------ otherwise | || 2 |
|| 2 | ||1 + cot (a) |
\\1 + cot (a) / \\ /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for And(im(a) = 0, 2*a mod pi = 0)\
// 1 for And(im(a) = 0, a mod pi = 0)\ || |
|| | || 1 |
1 - |< 1 | + |<------------- otherwise |
||-------- otherwise | || / pi\ |
\\sec(2*a) / ||sec|2*a - --| |
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// / / pi\ \\
|| 1 for And|im(a) = 0, |2*a + --| mod 2*pi = 0||
___ || \ \ 4 / /|
1 - \/ 2 *|< |
|| 2/ pi\ / 2/ pi\\ |
||sin |a + --|*|-1 + cot |a + --|| otherwise |
\\ \ 8 / \ \ 8 // /
$$\left(- \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(2 a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(a + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$
___
\/ 2
1 - --------------
/ pi\
csc|-2*a + --|
\ 4 /
$$1 - \frac{\sqrt{2}}{\csc{\left(- 2 a + \frac{\pi}{4} \right)}}$$
// 1 for And(im(a) = 0, a mod pi = 0)\ // 0 for And(im(a) = 0, 2*a mod pi = 0)\
|| | || |
||/ 1 for And(im(a) = 0, a mod pi = 0) | ||/ 0 for And(im(a) = 0, 2*a mod pi = 0) |
||| | ||| |
1 - |<| 2 | + |<| 2*cot(a) |
||<-1 + cot (a) otherwise | ||<----------- otherwise otherwise |
|||------------ otherwise | ||| 2 |
||| 2 | |||1 + cot (a) |
\\\1 + cot (a) / \\\ /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
1 1
1 + ------------- - --------
/ pi\ sec(2*a)
sec|2*a - --|
\ 2 /
$$1 + \frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(2 a \right)}}$$
/ pi\
1 - cos(2*a) + cos|2*a - --|
\ 2 /
$$- \cos{\left(2 a \right)} + \cos{\left(2 a - \frac{\pi}{2} \right)} + 1$$
2
1 - tan (a) 2*tan(a)
1 - ----------- + -----------
2 2
1 + tan (a) 1 + tan (a)
$$- \frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + 1 + \frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}$$
___ / 2/ pi\\
\/ 2 *|1 - tan |a + --||
\ \ 8 //
1 - ------------------------
2/ pi\
1 + tan |a + --|
\ 8 /
$$- \frac{\sqrt{2} \cdot \left(1 - \tan^{2}{\left(a + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1} + 1$$
___ / 3*pi\
1 - \/ 2 *sin|2*a + ----|
\ 4 /
$$- \sqrt{2} \sin{\left(2 a + \frac{3 \pi}{4} \right)} + 1$$
// 1 for And(im(a) = 0, a mod pi = 0)\
|| | // 0 for And(im(a) = 0, 2*a mod pi = 0)\
|| 1 | || |
1 - |<------------- otherwise | + |< 1 |
|| /pi \ | ||-------- otherwise |
||csc|-- - 2*a| | \\csc(2*a) /
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{1}{\csc{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for And(im(a) = 0, a mod pi = 0)\
|| | // 0 for And(im(a) = 0, 2*a mod pi = 0)\
1 - |< /pi \ | + |< |
||sin|-- + 2*a| otherwise | \\sin(2*a) otherwise /
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for And(im(a) = 0, 2*a mod pi = 0)\
// 1 for And(im(a) = 0, a mod pi = 0)\ || |
1 - |< | + |< / pi\ |
\\cos(2*a) otherwise / ||cos|2*a - --| otherwise |
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\cos{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for And(im(a) = 0, a mod pi = 0)\ // 0 for And(im(a) = 0, 2*a mod pi = 0)\
1 - |< | + |< |
\\cos(2*a) otherwise / \\sin(2*a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1$$
___ / pi\
1 - \/ 2 *cos|2*a + --|
\ 4 /
$$- \sqrt{2} \cos{\left(2 a + \frac{\pi}{4} \right)} + 1$$
// / / pi\ \\
|| 1 for And|im(a) = 0, |2*a + --| mod 2*pi = 0||
|| \ \ 4 / /|
|| |
___ || 2/ pi\ |
1 - \/ 2 *|<-1 + cot |a + --| |
|| \ 8 / |
||----------------- otherwise |
|| 2/ pi\ |
|| 1 + cot |a + --| |
\\ \ 8 / /
$$\left(- \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(2 a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(a + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(a + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$
/pi \
1 - sin|-- + 2*a| + sin(2*a)
\2 /
$$\sin{\left(2 a \right)} - \sin{\left(2 a + \frac{\pi}{2} \right)} + 1$$
1 1
1 + -------- - --------
csc(2*a) sec(2*a)
$$1 - \frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\csc{\left(2 a \right)}}$$
1 1
1 + -------- - -------------
csc(2*a) /pi \
csc|-- - 2*a|
\2 /
$$1 - \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(2 a \right)}}$$
// 1 for And(im(a) = 0, a mod pi = 0)\ // 0 for And(im(a) = 0, 2*a mod pi = 0)\
|| | || |
1 - | 1 for And(im(a) = 0, a mod pi = 0) | + | 0 for And(im(a) = 0, 2*a mod pi = 0) |
||< otherwise | ||< otherwise |
\\\cos(2*a) otherwise / \\\sin(2*a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
___
\/ 2
1 - -------------
/ pi\
sec|2*a + --|
\ 4 /
$$1 - \frac{\sqrt{2}}{\sec{\left(2 a + \frac{\pi}{4} \right)}}$$
// 1 for And(im(a) = 0, a mod pi = 0)\ // 0 for And(im(a) = 0, 2*a mod pi = 0)\
|| | || |
|| 2 | || 2*tan(a) |
1 - |<1 - tan (a) | + |<----------- otherwise |
||----------- otherwise | || 2 |
|| 2 | ||1 + tan (a) |
\\1 + tan (a) / \\ /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$