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