Господин Экзамен
Общий знаменатель 1+sin(3*pi/(2+a))/1+sin(pi/(2+a))
Решение
Вы ввели
[src]
/ 3*pi\
sin|-----|
\2 + a/ / pi \
1 + ---------- + sin|-----|
1 \2 + a/
$$\sin{\left(\frac{\pi}{a + 2} \right)} + \frac{\sin{\left(\frac{3 \pi}{a + 2} \right)}}{1} + 1$$
1 + sin(3*pi/(2 + a))/1 + sin(pi/(2 + a))
Общее упрощение
[src]
/ pi \ / 3*pi\
1 + sin|-----| + sin|-----|
\2 + a/ \2 + a/
$$\sin{\left(\frac{\pi}{a + 2} \right)} + \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1$$
1 + sin(pi/(2 + a)) + sin(3*pi/(2 + a))
Собрать выражение
[src]
/ pi \ / 3*pi\
1 + sin|-----| + sin|-----|
\2 + a/ \2 + a/
$$\sin{\left(\frac{\pi}{a + 2} \right)} + \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1$$
1 + sin(pi/(2 + a)) + sin(3*pi/(2 + a))
Комбинаторика
[src]
/ pi \ / 3*pi\
1 + sin|-----| + sin|-----|
\2 + a/ \2 + a/
$$\sin{\left(\frac{\pi}{a + 2} \right)} + \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1$$
1 + sin(pi/(2 + a)) + sin(3*pi/(2 + a))
Общий знаменатель
[src]
/ pi \ / 3*pi\
1 + sin|-----| + sin|-----|
\2 + a/ \2 + a/
$$\sin{\left(\frac{\pi}{a + 2} \right)} + \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1$$
1 + sin(pi/(2 + a)) + sin(3*pi/(2 + a))
Тригонометрическая часть
[src]
/ pi \ / 3*pi\
1 + sin|-----| + sin|-----|
\2 + a/ \2 + a/
$$\sin{\left(\frac{\pi}{a + 2} \right)} + \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1$$
/ pi \ / 2*pi\
1 + 2*cos|-----|*sin|-----|
\2 + a/ \2 + a/
$$2 \sin{\left(\frac{2 \pi}{a + 2} \right)} \cos{\left(\frac{\pi}{a + 2} \right)} + 1$$
1 1
1 + ---------- + ----------
/ pi \ / 3*pi\
csc|-----| csc|-----|
\2 + a/ \2 + a/
$$1 + \frac{1}{\csc{\left(\frac{3 \pi}{a + 2} \right)}} + \frac{1}{\csc{\left(\frac{\pi}{a + 2} \right)}}$$
1 1
1 + --------------- + ---------------
/ pi \ / 3*pi\
csc|pi - -----| csc|pi - -----|
\ 2 + a/ \ 2 + a/
$$1 + \frac{1}{\csc{\left(\pi - \frac{\pi}{a + 2} \right)}} + \frac{1}{\csc{\left(\pi - \frac{3 \pi}{a + 2} \right)}}$$
/ pi pi \ / pi 3*pi\
1 + cos|- -- + -----| + cos|- -- + -----|
\ 2 2 + a/ \ 2 2 + a/
$$\cos{\left(- \frac{\pi}{2} + \frac{\pi}{a + 2} \right)} + \cos{\left(- \frac{\pi}{2} + \frac{3 \pi}{a + 2} \right)} + 1$$
1 1
1 + --------------- + ---------------
/pi pi \ /pi 3*pi\
sec|-- - -----| sec|-- - -----|
\2 2 + a/ \2 2 + a/
$$1 + \frac{1}{\sec{\left(\frac{\pi}{2} - \frac{\pi}{a + 2} \right)}} + \frac{1}{\sec{\left(\frac{\pi}{2} - \frac{3 \pi}{a + 2} \right)}}$$
1 1
1 + ----------------- + -----------------
/ pi pi \ / pi 3*pi\
sec|- -- + -----| sec|- -- + -----|
\ 2 2 + a/ \ 2 2 + a/
$$1 + \frac{1}{\sec{\left(- \frac{\pi}{2} + \frac{3 \pi}{a + 2} \right)}} + \frac{1}{\sec{\left(- \frac{\pi}{2} + \frac{\pi}{a + 2} \right)}}$$
/ pi \ / 3*pi \
2*cot|---------| 2*cot|---------|
\2*(2 + a)/ \2*(2 + a)/
1 + ------------------- + -------------------
2/ pi \ 2/ 3*pi \
1 + cot |---------| 1 + cot |---------|
\2*(2 + a)/ \2*(2 + a)/
$$1 + \frac{2 \cot{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1} + \frac{2 \cot{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\cot^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)} + 1}$$
/ pi \ / 3*pi \
2*tan|---------| 2*tan|---------|
\2*(2 + a)/ \2*(2 + a)/
1 + ------------------- + -------------------
2/ pi \ 2/ 3*pi \
1 + tan |---------| 1 + tan |---------|
\2*(2 + a)/ \2*(2 + a)/
$$1 + \frac{2 \tan{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1} + \frac{2 \tan{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\tan^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)} + 1}$$
2 2
1 + ------------------------------------ + ------------------------------------
/ 1 \ / pi \ / 1 \ / 3*pi \
|1 + ---------------|*cot|---------| |1 + ---------------|*cot|---------|
| 2/ pi \| \2*(2 + a)/ | 2/ 3*pi \| \2*(2 + a)/
| cot |---------|| | cot |---------||
\ \2*(2 + a)// \ \2*(2 + a)//
$$1 + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}\right) \cot{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}\right) \cot{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}$$
/ / pi \\ / 2/pi pi \\ / / 3*pi\\ / 2/pi 3*pi \\
|1 - sin|-----||*|-1 + cot |-- - ---------|| |1 - sin|-----||*|-1 + cot |-- - ---------||
\ \2 + a// \ \4 2*(2 + a)// \ \2 + a// \ \4 2*(2 + a)//
1 + -------------------------------------------- + --------------------------------------------
2 2
$$\frac{\left(- \sin{\left(\frac{\pi}{a + 2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{\pi}{4} - \frac{\pi}{2 \left(a + 2\right)} \right)} - 1\right)}{2} + \frac{\left(- \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{\pi}{4} - \frac{3 \pi}{2 \left(a + 2\right)} \right)} - 1\right)}{2} + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
1 + |< | + |< |
|| / pi \ | || / 3*pi\ |
||sin|-----| otherwise | ||sin|-----| otherwise |
\\ \2 + a/ / \\ \2 + a/ /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\sin{\left(\frac{\pi}{a + 2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\sin{\left(\frac{3 \pi}{a + 2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
2/pi 3*pi \ 2/pi pi \
-1 + cot |-- - ---------| -1 + cot |-- - ---------|
\4 2*(2 + a)/ \4 2*(2 + a)/
1 + ------------------------- + -------------------------
2/pi 3*pi \ 2/pi pi \
1 + cot |-- - ---------| 1 + cot |-- - ---------|
\4 2*(2 + a)/ \4 2*(2 + a)/
$$\frac{\cot^{2}{\left(\frac{\pi}{4} - \frac{3 \pi}{2 \left(a + 2\right)} \right)} - 1}{\cot^{2}{\left(\frac{\pi}{4} - \frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1} + \frac{\cot^{2}{\left(\frac{\pi}{4} - \frac{\pi}{2 \left(a + 2\right)} \right)} - 1}{\cot^{2}{\left(\frac{\pi}{4} - \frac{\pi}{2 \left(a + 2\right)} \right)} + 1} + 1$$
2/pi 3*pi \ 2/pi pi \
1 - tan |-- - ---------| 1 - tan |-- - ---------|
\4 2*(2 + a)/ \4 2*(2 + a)/
1 + ------------------------ + ------------------------
2/pi 3*pi \ 2/pi pi \
1 + tan |-- - ---------| 1 + tan |-- - ---------|
\4 2*(2 + a)/ \4 2*(2 + a)/
$$\frac{- \tan^{2}{\left(\frac{\pi}{4} - \frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1}{\tan^{2}{\left(\frac{\pi}{4} - \frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1} + \frac{- \tan^{2}{\left(\frac{\pi}{4} - \frac{\pi}{2 \left(a + 2\right)} \right)} + 1}{\tan^{2}{\left(\frac{\pi}{4} - \frac{\pi}{2 \left(a + 2\right)} \right)} + 1} + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
1 + |< 1 | + |< 1 |
||---------- otherwise | ||---------- otherwise |
|| / pi \ | || / 3*pi\ |
||csc|-----| | ||csc|-----| |
\\ \2 + a/ / \\ \2 + a/ /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{1}{\csc{\left(\frac{\pi}{a + 2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{1}{\csc{\left(\frac{3 \pi}{a + 2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
1 + |< | + |< |
|| / pi pi \ | || / pi 3*pi\ |
||cos|- -- + -----| otherwise | ||cos|- -- + -----| otherwise |
\\ \ 2 2 + a/ / \\ \ 2 2 + a/ /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\cos{\left(- \frac{\pi}{2} + \frac{\pi}{a + 2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\cos{\left(- \frac{\pi}{2} + \frac{3 \pi}{a + 2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
1 + |< 1 | + |< 1 |
||----------------- otherwise | ||----------------- otherwise |
|| / pi pi \ | || / pi 3*pi\ |
||sec|- -- + -----| | ||sec|- -- + -----| |
\\ \ 2 2 + a/ / \\ \ 2 2 + a/ /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{1}{\sec{\left(- \frac{\pi}{2} + \frac{\pi}{a + 2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{1}{\sec{\left(- \frac{\pi}{2} + \frac{3 \pi}{a + 2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
2/ pi \ / pi \ 2/ 3*pi \ / 3*pi\
4*sin |---------|*sin|-----| 4*sin |---------|*sin|-----|
\2*(2 + a)/ \2 + a/ \2*(2 + a)/ \2 + a/
1 + ------------------------------- + -------------------------------
2/ pi \ 4/ pi \ 2/ 3*pi\ 4/ 3*pi \
sin |-----| + 4*sin |---------| sin |-----| + 4*sin |---------|
\2 + a/ \2*(2 + a)/ \2 + a/ \2*(2 + a)/
$$\frac{4 \sin^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} \sin{\left(\frac{3 \pi}{a + 2} \right)}}{4 \sin^{4}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + \sin^{2}{\left(\frac{3 \pi}{a + 2} \right)}} + \frac{4 \sin^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)} \sin{\left(\frac{\pi}{a + 2} \right)}}{4 \sin^{4}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)} + \sin^{2}{\left(\frac{\pi}{a + 2} \right)}} + 1$$
2/ pi \ 2/ 3*pi \
4*sin |---------| 4*sin |---------|
\2*(2 + a)/ \2*(2 + a)/
1 + ---------------------------------- + ----------------------------------
/ 4/ pi \\ / 4/ 3*pi \\
| 4*sin |---------|| | 4*sin |---------||
| \2*(2 + a)/| / pi \ | \2*(2 + a)/| / 3*pi\
|1 + -----------------|*sin|-----| |1 + -----------------|*sin|-----|
| 2/ pi \ | \2 + a/ | 2/ 3*pi\ | \2 + a/
| sin |-----| | | sin |-----| |
\ \2 + a/ / \ \2 + a/ /
$$1 + \frac{4 \sin^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\sin^{2}{\left(\frac{3 \pi}{a + 2} \right)}} + 1\right) \sin{\left(\frac{3 \pi}{a + 2} \right)}} + \frac{4 \sin^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\sin^{2}{\left(\frac{\pi}{a + 2} \right)}} + 1\right) \sin{\left(\frac{\pi}{a + 2} \right)}}$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
|| / pi \ | || / 3*pi \ |
1 + |< 2*cot|---------| | + |< 2*cot|---------| |
|| \2*(2 + a)/ | || \2*(2 + a)/ |
||------------------- otherwise | ||------------------- otherwise |
|| 2/ pi \ | || 2/ 3*pi \ |
||1 + cot |---------| | ||1 + cot |---------| |
\\ \2*(2 + a)/ / \\ \2*(2 + a)/ /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{2 \cot{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\cot^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{2 \cot{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
|| / pi \ | || / 3*pi \ |
1 + |< 2*tan|---------| | + |< 2*tan|---------| |
|| \2*(2 + a)/ | || \2*(2 + a)/ |
||------------------- otherwise | ||------------------- otherwise |
|| 2/ pi \ | || 2/ 3*pi \ |
||1 + tan |---------| | ||1 + tan |---------| |
\\ \2*(2 + a)/ / \\ \2*(2 + a)/ /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{2 \tan{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\tan^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{2 \tan{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
||/ / 1 \ | ||/ / 3 \ |
1 + |<| 0 for pi*|----- mod 1| = 0 | + |<| 0 for pi*|----- mod 1| = 0 |
||| \2 + a / | ||| \2 + a / |
||< otherwise | ||< otherwise |
||| / pi \ | ||| / 3*pi\ |
|||sin|-----| otherwise | |||sin|-----| otherwise |
\\\ \2 + a/ / \\\ \2 + a/ /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\sin{\left(\frac{\pi}{a + 2} \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\sin{\left(\frac{3 \pi}{a + 2} \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
|| 2 | || 2 |
1 + |<------------------------------------ otherwise | + |<------------------------------------ otherwise |
||/ 1 \ / pi \ | ||/ 1 \ / 3*pi \ |
|||1 + ---------------|*tan|---------| | |||1 + ---------------|*tan|---------| |
||| 2/ pi \| \2*(2 + a)/ | ||| 2/ 3*pi \| \2*(2 + a)/ |
||| tan |---------|| | ||| tan |---------|| |
\\\ \2*(2 + a)// / \\\ \2*(2 + a)// /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}\right) \tan{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}\right) \tan{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
/pi pi \ /pi 3*pi \
2*csc|-- - ---------| 2*csc|-- - ---------|
\2 2*(2 + a)/ \2 2*(2 + a)/
1 + ----------------------------------------- + -----------------------------------------
/ 2/pi pi \\ / 2/pi 3*pi \\
| csc |-- - ---------|| | csc |-- - ---------||
| \2 2*(2 + a)/| / pi \ | \2 2*(2 + a)/| / 3*pi \
|1 + --------------------|*csc|---------| |1 + --------------------|*csc|---------|
| 2/ pi \ | \2*(2 + a)/ | 2/ 3*pi \ | \2*(2 + a)/
| csc |---------| | | csc |---------| |
\ \2*(2 + a)/ / \ \2*(2 + a)/ /
$$1 + \frac{2 \csc{\left(\frac{\pi}{2} - \frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\left(1 + \frac{\csc^{2}{\left(\frac{\pi}{2} - \frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\csc^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}\right) \csc{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}} + \frac{2 \csc{\left(\frac{\pi}{2} - \frac{\pi}{2 \left(a + 2\right)} \right)}}{\left(1 + \frac{\csc^{2}{\left(\frac{\pi}{2} - \frac{\pi}{2 \left(a + 2\right)} \right)}}{\csc^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}\right) \csc{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}$$
/ pi \ / 3*pi \
2*sec|---------| 2*sec|---------|
\2*(2 + a)/ \2*(2 + a)/
1 + -------------------------------------------------- + --------------------------------------------------
/ 2/ pi \ \ / 2/ 3*pi \ \
| sec |---------| | | sec |---------| |
| \2*(2 + a)/ | / pi pi \ | \2*(2 + a)/ | / pi 3*pi \
|1 + ----------------------|*sec|- -- + ---------| |1 + ----------------------|*sec|- -- + ---------|
| 2/ pi pi \| \ 2 2*(2 + a)/ | 2/ pi 3*pi \| \ 2 2*(2 + a)/
| sec |- -- + ---------|| | sec |- -- + ---------||
\ \ 2 2*(2 + a)// \ \ 2 2*(2 + a)//
$$1 + \frac{2 \sec{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\left(\frac{\sec^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + \frac{3 \pi}{2 \left(a + 2\right)} \right)}} + 1\right) \sec{\left(- \frac{\pi}{2} + \frac{3 \pi}{2 \left(a + 2\right)} \right)}} + \frac{2 \sec{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\left(\frac{\sec^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(a + 2\right)} \right)}} + 1\right) \sec{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(a + 2\right)} \right)}}$$
/ pi pi \ / pi 3*pi \
2*cos|- -- + ---------| 2*cos|- -- + ---------|
\ 2 2*(2 + a)/ \ 2 2*(2 + a)/
1 + ------------------------------------------- + -------------------------------------------
/ 2/ pi pi \\ / 2/ pi 3*pi \\
| cos |- -- + ---------|| | cos |- -- + ---------||
| \ 2 2*(2 + a)/| / pi \ | \ 2 2*(2 + a)/| / 3*pi \
|1 + ----------------------|*cos|---------| |1 + ----------------------|*cos|---------|
| 2/ pi \ | \2*(2 + a)/ | 2/ 3*pi \ | \2*(2 + a)/
| cos |---------| | | cos |---------| |
\ \2*(2 + a)/ / \ \2*(2 + a)/ /
$$1 + \frac{2 \cos{\left(- \frac{\pi}{2} + \frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\left(1 + \frac{\cos^{2}{\left(- \frac{\pi}{2} + \frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\cos^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}\right) \cos{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}} + \frac{2 \cos{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(a + 2\right)} \right)}}{\left(1 + \frac{\cos^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(a + 2\right)} \right)}}{\cos^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}\right) \cos{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}$$
// //3 1 \ \ \ // //3 3 \ \ \
|| 1 for pi*||- + -----| mod 2| = 0| || 1 for pi*||- + -----| mod 2| = 0|
|| \\2 2 + a/ / | || \\2 2 + a/ / |
|| | || |
1 + |
$$\left(\begin{cases} 1 & \text{for}\: \pi \left(\left(\frac{3}{2} + \frac{1}{a + 2}\right) \bmod 2\right) = 0 \\\frac{\left(- \sin{\left(\frac{\pi}{a + 2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{\pi}{4} - \frac{\pi}{2 \left(a + 2\right)} \right)} - 1\right)}{2} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \pi \left(\left(\frac{3}{2} + \frac{3}{a + 2}\right) \bmod 2\right) = 0 \\\frac{\left(- \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{\pi}{4} - \frac{3 \pi}{2 \left(a + 2\right)} \right)} - 1\right)}{2} & \text{otherwise} \end{cases}\right) + 1$$
// //3 1 \ \ \ // //3 3 \ \ \
|| 1 for pi*||- + -----| mod 2| = 0| || 1 for pi*||- + -----| mod 2| = 0|
|| \\2 2 + a/ / | || \\2 2 + a/ / |
|| | || |
|| 2/pi pi \ | || 2/pi 3*pi \ |
1 + |<-1 + cot |-- - ---------| | + |<-1 + cot |-- - ---------| |
|| \4 2*(2 + a)/ | || \4 2*(2 + a)/ |
||------------------------- otherwise | ||------------------------- otherwise |
|| 2/pi pi \ | || 2/pi 3*pi \ |
|| 1 + cot |-- - ---------| | || 1 + cot |-- - ---------| |
\\ \4 2*(2 + a)/ / \\ \4 2*(2 + a)/ /
$$\left(\begin{cases} 1 & \text{for}\: \pi \left(\left(\frac{3}{2} + \frac{1}{a + 2}\right) \bmod 2\right) = 0 \\\frac{\cot^{2}{\left(\frac{\pi}{4} - \frac{\pi}{2 \left(a + 2\right)} \right)} - 1}{\cot^{2}{\left(\frac{\pi}{4} - \frac{\pi}{2 \left(a + 2\right)} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \pi \left(\left(\frac{3}{2} + \frac{3}{a + 2}\right) \bmod 2\right) = 0 \\\frac{\cot^{2}{\left(\frac{\pi}{4} - \frac{3 \pi}{2 \left(a + 2\right)} \right)} - 1}{\cot^{2}{\left(\frac{\pi}{4} - \frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
|| / pi \ | || / 3*pi\ |
|| sin|-----| | || sin|-----| |
|| \2 + a/ | || \2 + a/ |
1 + |<--------------------------------------- otherwise | + |<--------------------------------------- otherwise |
||/ 2/ pi \ \ | ||/ 2/ 3*pi\ \ |
||| sin |-----| | | ||| sin |-----| | |
||| \2 + a/ | 2/ pi \ | ||| \2 + a/ | 2/ 3*pi \ |
|||1 + -----------------|*sin |---------| | |||1 + -----------------|*sin |---------| |
||| 4/ pi \| \2*(2 + a)/ | ||| 4/ 3*pi \| \2*(2 + a)/ |
||| 4*sin |---------|| | ||| 4*sin |---------|| |
\\\ \2*(2 + a)// / \\\ \2*(2 + a)// /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{\sin{\left(\frac{\pi}{a + 2} \right)}}{\left(1 + \frac{\sin^{2}{\left(\frac{\pi}{a + 2} \right)}}{4 \sin^{4}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}\right) \sin^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{\sin{\left(\frac{3 \pi}{a + 2} \right)}}{\left(1 + \frac{\sin^{2}{\left(\frac{3 \pi}{a + 2} \right)}}{4 \sin^{4}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}\right) \sin^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
||/ / 1 \ | ||/ / 3 \ |
||| 0 for pi*|----- mod 1| = 0 | ||| 0 for pi*|----- mod 1| = 0 |
||| \2 + a / | ||| \2 + a / |
1 + |<| | + |<| |
||| / pi \ | ||| / 3*pi \ |
||< 2*cot|---------| otherwise | ||< 2*cot|---------| otherwise |
||| \2*(2 + a)/ | ||| \2*(2 + a)/ |
|||------------------- otherwise | |||------------------- otherwise |
||| 2/ pi \ | ||| 2/ 3*pi \ |
|||1 + cot |---------| | |||1 + cot |---------| |
\\\ \2*(2 + a)/ / \\\ \2*(2 + a)/ /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{2 \cot{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\cot^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{2 \cot{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
|| / pi \ | || / 3*pi \ |
|| 2*csc|---------| | || 2*csc|---------| |
|| \2*(2 + a)/ | || \2*(2 + a)/ |
1 + |<---------------------------------------------- otherwise | + |<---------------------------------------------- otherwise |
||/ 2/ pi \ \ | ||/ 2/ 3*pi \ \ |
||| csc |---------| | | ||| csc |---------| | |
||| \2*(2 + a)/ | /pi pi \ | ||| \2*(2 + a)/ | /pi 3*pi \ |
|||1 + --------------------|*csc|-- - ---------| | |||1 + --------------------|*csc|-- - ---------| |
||| 2/pi pi \| \2 2*(2 + a)/ | ||| 2/pi 3*pi \| \2 2*(2 + a)/ |
||| csc |-- - ---------|| | ||| csc |-- - ---------|| |
\\\ \2 2*(2 + a)// / \\\ \2 2*(2 + a)// /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{2 \csc{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\left(\frac{\csc^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\csc^{2}{\left(\frac{\pi}{2} - \frac{\pi}{2 \left(a + 2\right)} \right)}} + 1\right) \csc{\left(\frac{\pi}{2} - \frac{\pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{2 \csc{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\left(\frac{\csc^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\csc^{2}{\left(\frac{\pi}{2} - \frac{3 \pi}{2 \left(a + 2\right)} \right)}} + 1\right) \csc{\left(\frac{\pi}{2} - \frac{3 \pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
|| / pi pi \ | || / pi 3*pi \ |
|| 2*sec|- -- + ---------| | || 2*sec|- -- + ---------| |
|| \ 2 2*(2 + a)/ | || \ 2 2*(2 + a)/ |
1 + |<------------------------------------------- otherwise | + |<------------------------------------------- otherwise |
||/ 2/ pi pi \\ | ||/ 2/ pi 3*pi \\ |
||| sec |- -- + ---------|| | ||| sec |- -- + ---------|| |
||| \ 2 2*(2 + a)/| / pi \ | ||| \ 2 2*(2 + a)/| / 3*pi \ |
|||1 + ----------------------|*sec|---------| | |||1 + ----------------------|*sec|---------| |
||| 2/ pi \ | \2*(2 + a)/ | ||| 2/ 3*pi \ | \2*(2 + a)/ |
||| sec |---------| | | ||| sec |---------| | |
\\\ \2*(2 + a)/ / / \\\ \2*(2 + a)/ / /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{2 \sec{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(a + 2\right)} \right)}}{\left(1 + \frac{\sec^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(a + 2\right)} \right)}}{\sec^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}\right) \sec{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{2 \sec{\left(- \frac{\pi}{2} + \frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\left(1 + \frac{\sec^{2}{\left(- \frac{\pi}{2} + \frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\sec^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}\right) \sec{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// / 1 \ \ // / 3 \ \
|| 0 for pi*|----- mod 1| = 0| || 0 for pi*|----- mod 1| = 0|
|| \2 + a / | || \2 + a / |
|| | || |
|| / pi \ | || / 3*pi \ |
|| 2*cos|---------| | || 2*cos|---------| |
|| \2*(2 + a)/ | || \2*(2 + a)/ |
1 + |<-------------------------------------------------- otherwise | + |<-------------------------------------------------- otherwise |
||/ 2/ pi \ \ | ||/ 2/ 3*pi \ \ |
||| cos |---------| | | ||| cos |---------| | |
||| \2*(2 + a)/ | / pi pi \ | ||| \2*(2 + a)/ | / pi 3*pi \ |
|||1 + ----------------------|*cos|- -- + ---------| | |||1 + ----------------------|*cos|- -- + ---------| |
||| 2/ pi pi \| \ 2 2*(2 + a)/ | ||| 2/ pi 3*pi \| \ 2 2*(2 + a)/ |
||| cos |- -- + ---------|| | ||| cos |- -- + ---------|| |
\\\ \ 2 2*(2 + a)// / \\\ \ 2 2*(2 + a)// /
$$\left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{1}{a + 2} \bmod 1\right) = 0 \\\frac{2 \cos{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\left(\frac{\cos^{2}{\left(\frac{\pi}{2 \left(a + 2\right)} \right)}}{\cos^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(a + 2\right)} \right)}} + 1\right) \cos{\left(- \frac{\pi}{2} + \frac{\pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3}{a + 2} \bmod 1\right) = 0 \\\frac{2 \cos{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\left(\frac{\cos^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}}{\cos^{2}{\left(- \frac{\pi}{2} + \frac{3 \pi}{2 \left(a + 2\right)} \right)}} + 1\right) \cos{\left(- \frac{\pi}{2} + \frac{3 \pi}{2 \left(a + 2\right)} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
1 + Piecewise((0, pi*(Mod(1/(2 + a) = 1), 0)), (2*cos(pi/(2*(2 + a)))/((1 + cos(pi/(2*(2 + a)))^2/cos(-pi/2 + pi/(2*(2 + a)))^2)*cos(-pi/2 + pi/(2*(2 + a)))), True)) + Piecewise((0, pi*(Mod(3/(2 + a) = 1), 0)), (2*cos(3*pi/(2*(2 + a)))/((1 + cos(3*pi/(2*(2 + a)))^2/cos(-pi/2 + 3*pi/(2*(2 + a)))^2)*cos(-pi/2 + 3*pi/(2*(2 + a)))), True))
Численный ответ
[src]
1.0 + 1.0*sin(3*pi/(2 + a)) + sin(pi/(2 + a))
1.0 + 1.0*sin(3*pi/(2 + a)) + sin(pi/(2 + a))
Объединение рациональных выражений
[src]
/ pi \ / 3*pi\
1 + sin|-----| + sin|-----|
\2 + a/ \2 + a/
$$\sin{\left(\frac{\pi}{a + 2} \right)} + \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1$$
1 + sin(pi/(2 + a)) + sin(3*pi/(2 + a))
Степени
[src]
/ pi \ / 3*pi\
1 + sin|-----| + sin|-----|
\2 + a/ \2 + a/
$$\sin{\left(\frac{\pi}{a + 2} \right)} + \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1$$
/ -pi*I pi*I\ / -3*pi*I 3*pi*I\
| ------ -----| | ------- ------|
| 2 + a 2 + a| | 2 + a 2 + a |
I*\- e + e / I*\- e + e /
1 - ---------------------- - ------------------------
2 2
$$- \frac{i \left(e^{\frac{i \pi}{a + 2}} - e^{- \frac{i \pi}{a + 2}}\right)}{2} - \frac{i \left(e^{\frac{3 i \pi}{a + 2}} - e^{- \frac{3 i \pi}{a + 2}}\right)}{2} + 1$$
1 - i*(-exp(-pi*i/(2 + a)) + exp(pi*i/(2 + a)))/2 - i*(-exp(-3*pi*i/(2 + a)) + exp(3*pi*i/(2 + a)))/2
Рациональный знаменатель
[src]
/ pi \ / 3*pi\
1 + sin|-----| + sin|-----|
\2 + a/ \2 + a/
$$\sin{\left(\frac{\pi}{a + 2} \right)} + \sin{\left(\frac{3 \pi}{a + 2} \right)} + 1$$
1 + sin(pi/(2 + a)) + sin(3*pi/(2 + a))