Господин Экзамен
cos(2*a)+sin(2*a)-5 если a=4
Решение
Вы ввели
[src]
cos(2*a) + sin(2*a) - 5
$$\sin{\left(2 a \right)} + \cos{\left(2 a \right)} - 5$$
cos(2*a) + sin(2*a) - 1*5
Общее упрощение
[src]
___ / pi\
-5 + \/ 2 *sin|2*a + --|
\ 4 /
$$\sqrt{2} \sin{\left(2 a + \frac{\pi}{4} \right)} - 5$$
-5 + sqrt(2)*sin(2*a + pi/4)
Подстановка условия
[src]
cos(2*a) + sin(2*a) - 1*5 при a = 4
подставляем
cos(2*a) + sin(2*a) - 5
$$\sin{\left(2 a \right)} + \cos{\left(2 a \right)} - 5$$
___ / pi\
-5 + \/ 2 *sin|2*a + --|
\ 4 /
$$\sqrt{2} \sin{\left(2 a + \frac{\pi}{4} \right)} - 5$$
переменные
a = 4
$$a = 4$$
___ / pi\
-5 + \/ 2 *sin|2*(4) + --|
\ 4 /
$$\sqrt{2} \sin{\left(2 (4) + \frac{\pi}{4} \right)} - 5$$
___ / pi\
-5 + \/ 2 *sin|2*4 + --|
\ 4 /
$$-5 + \sqrt{2} \sin{\left(\frac{\pi}{4} + 2 \cdot 4 \right)}$$
___ / pi\
-5 + \/ 2 *sin|8 + --|
\ 4 /
$$-5 + \sqrt{2} \sin{\left(\frac{\pi}{4} + 8 \right)}$$
-5 + sqrt(2)*sin(8 + pi/4)
Тригонометрическая часть
[src]
___ / pi\
-5 + \/ 2 *cos|2*a - --|
\ 4 /
$$\sqrt{2} \cos{\left(2 a - \frac{\pi}{4} \right)} - 5$$
___ / pi\
-5 + \/ 2 *sin|2*a + --|
\ 4 /
$$\sqrt{2} \sin{\left(2 a + \frac{\pi}{4} \right)} - 5$$
2 -6 + 2*cos (a) + sin(2*a)
$$2 \cos^{2}{\left(a \right)} + \sin{\left(2 a \right)} - 6$$
___
\/ 2
-5 + -------------
/ pi\
sec|2*a - --|
\ 4 /
$$-5 + \frac{\sqrt{2}}{\sec{\left(2 a - \frac{\pi}{4} \right)}}$$
___
\/ 2
-5 + -------------
/ pi\
csc|2*a + --|
\ 4 /
$$-5 + \frac{\sqrt{2}}{\csc{\left(2 a + \frac{\pi}{4} \right)}}$$
/pi \
-5 + sin(2*a) + sin|-- + 2*a|
\2 /
$$\sin{\left(2 a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)} - 5$$
/ pi\
-5 + cos(2*a) + cos|2*a - --|
\ 2 /
$$\cos{\left(2 a \right)} + \cos{\left(2 a - \frac{\pi}{2} \right)} - 5$$
1 1
-5 + -------- + --------
csc(2*a) sec(2*a)
$$-5 + \frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\csc{\left(2 a \right)}}$$
1 1
-5 + -------- + -------------
sec(2*a) / pi\
sec|2*a - --|
\ 2 /
$$-5 + \frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}}$$
1 1
-5 + -------- + -------------
sec(2*a) /pi \
sec|-- - 2*a|
\2 /
$$-5 + \frac{1}{\sec{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}}$$
1 1
-5 + -------- + -------------
csc(2*a) /pi \
csc|-- - 2*a|
\2 /
$$-5 + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(2 a \right)}}$$
1 1
-5 + ------------- + -------------
csc(pi - 2*a) /pi \
csc|-- - 2*a|
\2 /
$$-5 + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 2 a + \pi \right)}}$$
2 2 -5 + cos (a) - sin (a) + 2*cos(a)*sin(a)
$$- \sin^{2}{\left(a \right)} + 2 \sin{\left(a \right)} \cos{\left(a \right)} + \cos^{2}{\left(a \right)} - 5$$
___ / pi\
2*\/ 2 *tan|a + --|
\ 8 /
-5 + -------------------
2/ pi\
1 + tan |a + --|
\ 8 /
$$-5 + \frac{2 \sqrt{2} \tan{\left(a + \frac{\pi}{8} \right)}}{\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1}$$
2
1 - tan (a) 2*tan(a)
-5 + ----------- + -----------
2 2
1 + tan (a) 1 + tan (a)
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} - 5 + \frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}$$
/ pi\
2*tan|a + --|
2*cot(a) \ 4 /
-5 + ----------- + ----------------
2 2/ pi\
1 + cot (a) 1 + tan |a + --|
\ 4 /
$$-5 + \frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} + \frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}$$
/ pi\
2*tan|a + --|
2*tan(a) \ 4 /
-5 + ----------- + ----------------
2 2/ pi\
1 + tan (a) 1 + tan |a + --|
\ 4 /
$$-5 + \frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}$$
2/ pi\
2 -1 + tan |a + --|
-1 + cot (a) \ 4 /
-5 + ------------ + -----------------
2 2/ pi\
1 + cot (a) 1 + tan |a + --|
\ 4 /
$$\frac{\tan^{2}{\left(a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} + \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} - 5$$
2/ pi\
1 - cot |a + --| 2
\ 4 / 1 - tan (a)
-5 + ---------------- + -----------
2/ pi\ 2
1 + cot |a + --| 1 + tan (a)
\ 4 /
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + \frac{- \cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} - 5$$
1
1 - -------
2
cot (a) 2
-5 + ----------- + --------------------
1 / 1 \
1 + ------- |1 + -------|*cot(a)
2 | 2 |
cot (a) \ cot (a)/
$$\frac{1 - \frac{1}{\cot^{2}{\left(a \right)}}}{1 + \frac{1}{\cot^{2}{\left(a \right)}}} - 5 + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(a \right)}}\right) \cot{\left(a \right)}}$$
// / pi\ \
|| 0 for |2*a + --| mod pi = 0|
___ || \ 4 / |
-5 + \/ 2 *|< |
|| 2/ pi\ / pi\ |
||2*sin |a + --|*cot|a + --| otherwise |
\\ \ 8 / \ 8 / /
$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(a + \frac{\pi}{8} \right)} \cot{\left(a + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right) - 5$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
-5 + |< | + |< |
\\sin(2*a) otherwise / \\cos(2*a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - 5$$
// / pi\ \
|| 0 for |2*a + --| mod pi = 0|
|| \ 4 / |
|| |
___ || / pi\ |
-5 + \/ 2 *|< 2*cot|a + --| |
|| \ 8 / |
||---------------- otherwise |
|| 2/ pi\ |
||1 + cot |a + --| |
\\ \ 8 / /
$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{8} \right)}}{\cot^{2}{\left(a + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - 5$$
// 0 for 2*a mod pi = 0\
|| | // 1 for a mod pi = 0\
-5 + |< / pi\ | + |< |
||cos|2*a - --| otherwise | \\cos(2*a) otherwise /
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\cos{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - 5$$
// 1 for a mod pi = 0\
// 0 for 2*a mod pi = 0\ || |
-5 + |< | + |< /pi \ |
\\sin(2*a) otherwise / ||sin|-- + 2*a| otherwise |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - 5$$
// /pi \ \
// 0 for 2*a mod pi = 0\ || 0 for |-- + 2*a| mod pi = 0|
-5 + |< | + |< \2 / |
\\sin(2*a) otherwise / || |
\\cos(2*a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - 5$$
// / 3*pi\ \
// 1 for a mod pi = 0\ || 1 for |2*a + ----| mod 2*pi = 0|
-5 + |< | + |< \ 2 / |
\\cos(2*a) otherwise / || |
\\sin(2*a) otherwise /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(2 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - 5$$
// 0 for 2*a mod pi = 0\
|| | // 1 for a mod pi = 0\
|| 1 | || |
-5 + |<------------- otherwise | + |< 1 |
|| / pi\ | ||-------- otherwise |
||sec|2*a - --| | \\sec(2*a) /
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) - 5$$
// 1 for a mod pi = 0\
// 0 for 2*a mod pi = 0\ || |
|| | || 1 |
-5 + |< 1 | + |<------------- otherwise |
||-------- otherwise | || /pi \ |
\\csc(2*a) / ||csc|-- - 2*a| |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{1}{\csc{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 5$$
4
4*sin (a)
1 - ---------
2 2
sin (2*a) 4*sin (a)
-5 + ------------- + ------------------------
4 / 4 \
4*sin (a) | 4*sin (a)|
1 + --------- |1 + ---------|*sin(2*a)
2 | 2 |
sin (2*a) \ sin (2*a)/
$$\frac{- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} - 5 + \frac{4 \sin^{2}{\left(a \right)}}{\left(\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1\right) \sin{\left(2 a \right)}}$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2*cot(a) | || 2 |
-5 + |<----------- otherwise | + |<-1 + cot (a) |
|| 2 | ||------------ otherwise |
||1 + cot (a) | || 2 |
\\ / \\1 + cot (a) /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - 5$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2*tan(a) | || 2 |
-5 + |<----------- otherwise | + |<1 - tan (a) |
|| 2 | ||----------- otherwise |
||1 + tan (a) | || 2 |
\\ / \\1 + tan (a) /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - 5$$
2/ pi\
cos |a - --|
\ 2 /
1 - ------------ / pi\
2 2*cos|a - --|
cos (a) \ 2 /
-5 + ---------------- + -------------------------
2/ pi\ / 2/ pi\\
cos |a - --| | cos |a - --||
\ 2 / | \ 2 /|
1 + ------------ |1 + ------------|*cos(a)
2 | 2 |
cos (a) \ cos (a) /
$$\frac{1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}{1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}} - 5 + \frac{2 \cos{\left(a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}\right) \cos{\left(a \right)}}$$
2
sec (a)
1 - ------------
2/ pi\
sec |a - --|
\ 2 / 2*sec(a)
-5 + ---------------- + ------------------------------
2 / 2 \
sec (a) | sec (a) | / pi\
1 + ------------ |1 + ------------|*sec|a - --|
2/ pi\ | 2/ pi\| \ 2 /
sec |a - --| | sec |a - --||
\ 2 / \ \ 2 //
$$\frac{- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} - 5 + \frac{2 \sec{\left(a \right)}}{\left(\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
2/pi \
csc |-- - a|
\2 /
1 - ------------ /pi \
2 2*csc|-- - a|
csc (a) \2 /
-5 + ---------------- + -------------------------
2/pi \ / 2/pi \\
csc |-- - a| | csc |-- - a||
\2 / | \2 /|
1 + ------------ |1 + ------------|*csc(a)
2 | 2 |
csc (a) \ csc (a) /
$$\frac{1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}}{1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}} - 5 + \frac{2 \csc{\left(- a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right) \csc{\left(a \right)}}$$
// 1 for a mod pi = 0\
// 0 for 2*a mod pi = 0\ || |
|| | || 1 |
|| 2 | ||-1 + ------- |
||-------------------- otherwise | || 2 |
-5 + |
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \tan{\left(a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(a \right)}}}{1 + \frac{1}{\tan^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) - 5$$
// /pi \ \
|| 0 for |-- + 2*a| mod pi = 0|
// 0 for 2*a mod pi = 0\ || \2 / |
|| | || |
|| 2*cot(a) | || / pi\ |
-5 + |<----------- otherwise | + |< 2*cot|a + --| |
|| 2 | || \ 4 / |
||1 + cot (a) | ||---------------- otherwise |
\\ / || 2/ pi\ |
||1 + cot |a + --| |
\\ \ 4 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) - 5$$
// / 3*pi\ \
|| 1 for |2*a + ----| mod 2*pi = 0|
// 1 for a mod pi = 0\ || \ 2 / |
|| | || |
|| 2 | || 2/ pi\ |
-5 + |<-1 + cot (a) | + |<-1 + tan |a + --| |
||------------ otherwise | || \ 4 / |
|| 2 | ||----------------- otherwise |
\\1 + cot (a) / || 2/ pi\ |
|| 1 + tan |a + --| |
\\ \ 4 / /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(2 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) - 5$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
-5 + |
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 5$$
// 1 for a mod pi = 0\
|| |
// 0 for 2*a mod pi = 0\ || 2 |
|| | || sin (2*a) |
|| sin(2*a) | ||-1 + --------- |
||----------------------- otherwise | || 4 |
-5 + |
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\sin{\left(2 a \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) - 5$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
||/ 0 for 2*a mod pi = 0 | ||/ 1 for a mod pi = 0 |
||| | ||| |
-5 + |<| 2*cot(a) | + |<| 2 |
||<----------- otherwise otherwise | ||<-1 + cot (a) otherwise |
||| 2 | |||------------ otherwise |
|||1 + cot (a) | ||| 2 |
\\\ / \\\1 + cot (a) /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 5$$
// 1 for a mod pi = 0\
|| |
// 0 for 2*a mod pi = 0\ || 2 |
|| | || cos (a) |
|| 2*cos(a) | ||-1 + ------------ |
||------------------------------ otherwise | || 2/ pi\ |
||/ 2 \ | || cos |a - --| |
-5 + |<| cos (a) | / pi\ | + |< \ 2 / |
|||1 + ------------|*cos|a - --| | ||----------------- otherwise |
||| 2/ pi\| \ 2 / | || 2 |
||| cos |a - --|| | || cos (a) |
||\ \ 2 // | || 1 + ------------ |
\\ / || 2/ pi\ |
|| cos |a - --| |
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cos{\left(a \right)}}{\left(\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - 5$$
// 1 for a mod pi = 0\
// 0 for 2*a mod pi = 0\ || |
|| | || 2/ pi\ |
|| / pi\ | || sec |a - --| |
|| 2*sec|a - --| | || \ 2 / |
|| \ 2 / | ||-1 + ------------ |
||------------------------- otherwise | || 2 |
-5 + |
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \sec{\left(a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}}{1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) - 5$$
// 1 for a mod pi = 0\
|| |
// 0 for 2*a mod pi = 0\ || 2 |
|| | || csc (a) |
|| 2*csc(a) | ||-1 + ------------ |
||------------------------------ otherwise | || 2/pi \ |
||/ 2 \ | || csc |-- - a| |
-5 + |<| csc (a) | /pi \ | + |< \2 / |
|||1 + ------------|*csc|-- - a| | ||----------------- otherwise |
||| 2/pi \| \2 / | || 2 |
||| csc |-- - a|| | || csc (a) |
||\ \2 // | || 1 + ------------ |
\\ / || 2/pi \ |
|| csc |-- - a| |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \csc{\left(a \right)}}{\left(\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - 5$$
-5 + Piecewise((0, Mod(2*a = pi, 0)), (2*csc(a)/((1 + csc(a)^2/csc(pi/2 - a)^2)*csc(pi/2 - a)), True)) + Piecewise((1, Mod(a = pi, 0)), ((-1 + csc(a)^2/csc(pi/2 - a)^2)/(1 + csc(a)^2/csc(pi/2 - a)^2), True))
Степени
[src]
-2*I*a 2*I*a / -2*I*a 2*I*a\
e e I*\- e + e /
-5 + ------- + ------ - ----------------------
2 2 2
$$\frac{e^{2 i a}}{2} - \frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2} - 5 + \frac{e^{- 2 i a}}{2}$$
-5 + exp(-2*i*a)/2 + exp(2*i*a)/2 - i*(-exp(-2*i*a) + exp(2*i*a))/2