Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sqrt(4-4*sin(2*a))*sqrt(2-2*cos(2*a)) если a=-1/3
- (7*p+2*q)^2+(7*p-2*q)^2 если q=-4
- 2*sin(6*x)*cos(6*x) если x=1/4
- log(a*b)^2 если a=-1/3
- Разложить многочлен на множители:
- x^2-11*x+24
- m^3+8
- 126*x^3+3*x^2+3*x+1
- 144-a^2
- Общий знаменатель:
- (u^2-2*u+4)/(4*u^2-1)*(2*u^2+4)/(u^3+8)
- t/(t+3)^3-3/(t+3)^3
- x^2-49/x^2+9*x*x^2-81/x^2-7*x
- sqrt(2)/(sqrt(8)+2)-((6-sqrt(32))/sqrt(8)-3)
- Умножение столбиком:
- 419*217
- 28*34
- 12*18
- 52*14
- Деление столбиком:
- 1000/9
- 15/7
- 592/6
- 1219/3
- Раскрыть скобки в:
- (3*x-11)*2-5*(4-3*x)
- 7*(a+b)
- 26*(-a+b-c)
- (a+2)*(a^2+2*a+4)
- Разложение числа на простые множители:
- 31024
- 388
- 4860
- 6652
- Производная:
- -1/3
- Интеграл d{x}:
-
-1/3
-
Идентичные выражения
- sqrt(четыре - четыре *sin(два *a))*sqrt(два - два *cos(два *a)) если a=- один / три
- квадратный корень из (4 минус 4 умножить на синус от (2 умножить на a)) умножить на квадратный корень из (2 минус 2 умножить на косинус от (2 умножить на a)) если a равно минус 1 делить на 3
- квадратный корень из (четыре минус четыре умножить на синус от (два умножить на a)) умножить на квадратный корень из (два минус два умножить на косинус от (два умножить на a)) если a равно минус один делить на три
- √(4-4*sin(2*a))*√(2-2*cos(2*a)) если a=-1/3
- sqrt(4-4sin(2a))sqrt(2-2cos(2a)) если a=-1/3
- sqrt4-4sin2asqrt2-2cos2a если a=-1/3
- sqrt(4-4*sin(2*a))*sqrt(2-2*cos(2*a)) при a=-1/3
- sqrt(4-4*sin(2*a))*sqrt(2-2*cos(2*a)) если a=-1 разделить на 3
-
Похожие выражения
- sqrt(4-4*sin(2*a))*sqrt(2-2*cos(2*a)) если a=+1/3
- sqrt(4-4*sin(2*a))*sqrt(2+2*cos(2*a)) если a=-1/3
- sqrt(4+4*sin(2*a))*sqrt(2-2*cos(2*a)) если a=-1/3
sqrt(4-4*sin(2*a))*sqrt(2-2*cos(2*a)) если a=-1/3
Решение
Вы ввели
[src]
________________ ________________ \/ 4 - 4*sin(2*a) *\/ 2 - 2*cos(2*a)
$$\sqrt{- 2 \cos{\left(2 a \right)} + 2} \sqrt{- 4 \sin{\left(2 a \right)} + 4}$$
sqrt(4 - 4*sin(2*a))*sqrt(2 - 2*cos(2*a))
Общее упрощение
[src]
_________
/ 2 ______________
4*\/ sin (a) *\/ 1 - sin(2*a)
$$4 \sqrt{- \sin{\left(2 a \right)} + 1} \sqrt{\sin^{2}{\left(a \right)}}$$
4*sqrt(sin(a)^2)*sqrt(1 - sin(2*a))
Подстановка условия
[src]
sqrt(4 - 4*sin(2*a))*sqrt(2 - 2*cos(2*a)) при a = -1/3
подставляем
________________ ________________ \/ 4 - 4*sin(2*a) *\/ 2 - 2*cos(2*a)
$$\sqrt{- 2 \cos{\left(2 a \right)} + 2} \sqrt{- 4 \sin{\left(2 a \right)} + 4}$$
_________
/ 2 ______________
4*\/ sin (a) *\/ 1 - sin(2*a)
$$4 \sqrt{- \sin{\left(2 a \right)} + 1} \sqrt{\sin^{2}{\left(a \right)}}$$
переменные
a = -1/3
$$a = - \frac{1}{3}$$
______________
/ 2 ___________________
4*\/ sin ((-1/3)) *\/ 1 - sin(2*(-1/3))
$$4 \sqrt{- \sin{\left(2 (-1/3) \right)} + 1} \sqrt{\sin^{2}{\left((-1/3) \right)}}$$
____________
/ 2 _________________
4*\/ sin (-1/3) *\/ 1 - sin(2*-1/3)
$$4 \sqrt{- \sin{\left(2 \left(- \frac{1}{3}\right) \right)} + 1} \sqrt{\sin^{2}{\left(- \frac{1}{3} \right)}}$$
______________ 4*\/ 1 + sin(2/3) *sin(1/3)
$$4 \sqrt{\sin{\left(\frac{2}{3} \right)} + 1} \sin{\left(\frac{1}{3} \right)}$$
4*sqrt(1 + sin(2/3))*sin(1/3)
Тригонометрическая часть
[src]
________________ 2*\/ 4 - 4*sin(2*a) *|sin(a)|
$$2 \sqrt{- 4 \sin{\left(2 a \right)} + 4} \left|{\sin{\left(a \right)}}\right|$$
______________ ______________ / 2 / 4 / 2 - -------- * / 4 - -------- \/ sec(2*a) \/ csc(2*a)
$$\sqrt{2 - \frac{2}{\sec{\left(2 a \right)}}} \sqrt{4 - \frac{4}{\csc{\left(2 a \right)}}}$$
______________
/ 4
2* / 4 - --------
\/ csc(2*a)
--------------------
|csc(a)|
$$\frac{2 \sqrt{4 - \frac{4}{\csc{\left(2 a \right)}}}}{\left|{\csc{\left(a \right)}}\right|}$$
_____________________ / /pi \ ________________ / 2 - 2*sin|-- + 2*a| *\/ 4 - 4*sin(2*a) \/ \2 /
$$\sqrt{- 2 \sin{\left(2 a + \frac{\pi}{2} \right)} + 2} \sqrt{- 4 \sin{\left(2 a \right)} + 4}$$
_____________________
________________ / / pi\
\/ 2 - 2*cos(2*a) * / 4 - 4*cos|2*a - --|
\/ \ 2 /
$$\sqrt{- 2 \cos{\left(2 a \right)} + 2} \sqrt{- 4 \cos{\left(2 a - \frac{\pi}{2} \right)} + 4}$$
______________ ___________________
/ 2 / 4
/ 2 - -------- * / 4 - -------------
\/ sec(2*a) / / pi\
/ sec|2*a - --|
\/ \ 2 /
$$\sqrt{2 - \frac{2}{\sec{\left(2 a \right)}}} \sqrt{4 - \frac{4}{\sec{\left(2 a - \frac{\pi}{2} \right)}}}$$
___________________ ______________
/ 2 / 4
/ 2 - ------------- * / 4 - --------
/ /pi \ \/ csc(2*a)
/ csc|-- - 2*a|
\/ \2 /
$$\sqrt{2 - \frac{2}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}} \sqrt{4 - \frac{4}{\csc{\left(2 a \right)}}}$$
______________ ___________________
/ 2 / 4
/ 2 - -------- * / 4 - -------------
\/ sec(2*a) / /pi \
/ sec|-- - 2*a|
\/ \2 /
$$\sqrt{2 - \frac{2}{\sec{\left(2 a \right)}}} \sqrt{4 - \frac{4}{\sec{\left(- 2 a + \frac{\pi}{2} \right)}}}$$
_____________________
/ / pi\ | / pi\|
2* / 4 - 4*cos|2*a - --| *|cos|a - --||
\/ \ 2 / | \ 2 /|
$$2 \sqrt{- 4 \cos{\left(2 a - \frac{\pi}{2} \right)} + 4} \left|{\cos{\left(a - \frac{\pi}{2} \right)}}\right|$$
___________________ ___________________
/ 2 / 4
/ 2 - ------------- * / 4 - -------------
/ /pi \ \/ csc(pi - 2*a)
/ csc|-- - 2*a|
\/ \2 /
$$\sqrt{2 - \frac{2}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}} \sqrt{4 - \frac{4}{\csc{\left(- 2 a + \pi \right)}}}$$
___________________
/ 4
2* / 4 - -------------
/ / pi\
/ sec|2*a - --|
\/ \ 2 /
---------------------------
| / pi\|
|sec|a - --||
| \ 2 /|
$$\frac{2 \sqrt{4 - \frac{4}{\sec{\left(2 a - \frac{\pi}{2} \right)}}}}{\left|{\sec{\left(a - \frac{\pi}{2} \right)}}\right|}$$
______________ | /a\|
4*\/ 1 - sin(2*a) *(1 + cos(a))*|tan|-||
| \2/|
$$4 \sqrt{- \sin{\left(2 a \right)} + 1} \left(\cos{\left(a \right)} + 1\right) \left|{\tan{\left(\frac{a}{2} \right)}}\right|$$
___________________________ _____________________ / 2 2 \/ 4 - 8*cos(a)*sin(a) *\/ 2 - 2*cos (a) + 2*sin (a)
$$\sqrt{- 8 \sin{\left(a \right)} \cos{\left(a \right)} + 4} \sqrt{2 \sin^{2}{\left(a \right)} - 2 \cos^{2}{\left(a \right)} + 2}$$
_________________
/ 8*tan(a) | /a\|
4* / 4 - ----------- *|tan|-||
/ 2 | \2/|
\/ 1 + tan (a)
---------------------------------
2/a\
1 + tan |-|
\2/
$$\frac{4 \sqrt{4 - \frac{8 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}} \left|{\tan{\left(\frac{a}{2} \right)}}\right|}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
_____________________
/ / 2 \ _________________
/ 2*\1 - tan (a)/ / 8*tan(a)
/ 2 - --------------- * / 4 - -----------
/ 2 / 2
\/ 1 + tan (a) \/ 1 + tan (a)
$$\sqrt{4 - \frac{8 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}} \sqrt{- \frac{2 \cdot \left(- \tan^{2}{\left(a \right)} + 1\right)}{\tan^{2}{\left(a \right)} + 1} + 2}$$
______________________
/ / pi\
/ 4*tan|a + --| _________________
/ \ 4 / / 8*tan(a)
/ 2 - ---------------- * / 4 - -----------
/ 2/ pi\ / 2
/ 1 + tan |a + --| \/ 1 + tan (a)
\/ \ 4 /
$$\sqrt{2 - \frac{4 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}} \sqrt{4 - \frac{8 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}}$$
______________________
/ / pi\
/ 4*tan|a + --| _________________
/ \ 4 / / 8*cot(a)
/ 2 - ---------------- * / 4 - -----------
/ 2/ pi\ / 2
/ 1 + tan |a + --| \/ 1 + cot (a)
\/ \ 4 /
$$\sqrt{2 - \frac{4 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}} \sqrt{4 - \frac{8 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}}$$
_____________________
/ / 1 \
/ 2*|1 - -------|
/ | 2 | __________________________
/ \ cot (a)/ / 8
/ 2 - --------------- * / 4 - --------------------
/ 1 / / 1 \
/ 1 + ------- / |1 + -------|*cot(a)
/ 2 / | 2 |
\/ cot (a) \/ \ cot (a)/
$$\sqrt{4 - \frac{8}{\left(1 + \frac{1}{\cot^{2}{\left(a \right)}}\right) \cot{\left(a \right)}}} \sqrt{- \frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(a \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(a \right)}}} + 2}$$
___________________________
______________________ / / 2/ pi\\
/ / 2 \ / 4*|-1 + tan |a + --||
/ 2*\-1 + cot (a)/ / \ \ 4 //
/ 2 - ---------------- * / 4 - ---------------------
/ 2 / 2/ pi\
\/ 1 + cot (a) / 1 + tan |a + --|
\/ \ 4 /
$$\sqrt{- \frac{4 \left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} + 4} \sqrt{- \frac{2 \left(\cot^{2}{\left(a \right)} - 1\right)}{\cot^{2}{\left(a \right)} + 1} + 2}$$
__________________________
_____________________ / / 2/ pi\\
/ / 2 \ / 4*|1 - cot |a + --||
/ 2*\1 - tan (a)/ / \ \ 4 //
/ 2 - --------------- * / 4 - --------------------
/ 2 / 2/ pi\
\/ 1 + tan (a) / 1 + cot |a + --|
\/ \ 4 /
$$\sqrt{- \frac{2 \cdot \left(- \tan^{2}{\left(a \right)} + 1\right)}{\tan^{2}{\left(a \right)} + 1} + 2} \sqrt{- \frac{4 \cdot \left(- \cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} + 4}$$
_______________________________________
/ // 0 for 2*a mod pi = 0\ |/ 0 for a mod pi = 0|
2* / 4 - 4*|< | *|< |
\/ \\sin(2*a) otherwise / |\sin(a) otherwise |
$$2 \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + 4}\right) \left(\left|{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}\right|\right)$$
_____________________________________ _______________________________________ / // 1 for a mod pi = 0\ / // 0 for 2*a mod pi = 0\ / 2 - 2*|< | * / 4 - 4*|< | \/ \\cos(2*a) otherwise / \/ \\sin(2*a) otherwise /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
_______________________
/ / 4 \
/ | 4*sin (a)|
/ 2*|1 - ---------| ______________________________
/ | 2 | / 2
/ \ sin (2*a)/ / 16*sin (a)
/ 2 - ----------------- * / 4 - ------------------------
/ 4 / / 4 \
/ 4*sin (a) / | 4*sin (a)|
/ 1 + --------- / |1 + ---------|*sin(2*a)
/ 2 / | 2 |
\/ sin (2*a) \/ \ sin (2*a)/
$$\sqrt{4 - \frac{16 \sin^{2}{\left(a \right)}}{\left(\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1\right) \sin{\left(2 a \right)}}} \sqrt{- \frac{2 \left(- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} + 2}$$
__________________________________________
/ // 1 for a mod pi = 0\ _______________________________________
/ || | / // 0 for 2*a mod pi = 0\
/ 2 - 2*|< /pi \ | * / 4 - 4*|< |
/ ||sin|-- + 2*a| otherwise | \/ \\sin(2*a) otherwise /
\/ \\ \2 / /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
____________________________________________
_____________________________________ / // 0 for 2*a mod pi = 0\
/ // 1 for a mod pi = 0\ / || |
/ 2 - 2*|< | * / 4 - 4*|< / pi\ |
\/ \\cos(2*a) otherwise / / ||cos|2*a - --| otherwise |
\/ \\ \ 2 / /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\cos{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
______________________________________________
/ // /pi \ \ _______________________________________
/ || 0 for |-- + 2*a| mod pi = 0| / // 0 for 2*a mod pi = 0\
/ 2 - 2*|< \2 / | * / 4 - 4*|< |
/ || | \/ \\sin(2*a) otherwise /
\/ \\cos(2*a) otherwise /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
__________________________________________________
_____________________________________ / // / 3*pi\ \
/ // 1 for a mod pi = 0\ / || 1 for |2*a + ----| mod 2*pi = 0|
/ 2 - 2*|< | * / 4 - 4*|< \ 2 / |
\/ \\cos(2*a) otherwise / / || |
\/ \\sin(2*a) otherwise /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 1 & \text{for}\: \left(2 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
____________________________________________
_____________________________________ / // 0 for 2*a mod pi = 0\
/ // 1 for a mod pi = 0\ / || |
/ || | / || 1 |
/ 2 - 2*|< 1 | * / 4 - 4*|<------------- otherwise |
/ ||-------- otherwise | / || / pi\ |
\/ \\sec(2*a) / / ||sec|2*a - --| |
\/ \\ \ 2 / /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
__________________________________________
/ // 1 for a mod pi = 0\ _______________________________________
/ || | / // 0 for 2*a mod pi = 0\
/ || 1 | / || |
/ 2 - 2*|<------------- otherwise | * / 4 - 4*|< 1 |
/ || /pi \ | / ||-------- otherwise |
/ ||csc|-- - 2*a| | \/ \\csc(2*a) /
\/ \\ \2 / /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{1}{\csc{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
__________________________
/ / 2 \
/ | sec (a) |
/ 2*|1 - ------------|
/ | 2/ pi\|
/ | sec |a - --|| ____________________________________
/ \ \ 2 // / 8*sec(a)
/ 2 - -------------------- * / 4 - ------------------------------
/ 2 / / 2 \
/ sec (a) / | sec (a) | / pi\
/ 1 + ------------ / |1 + ------------|*sec|a - --|
/ 2/ pi\ / | 2/ pi\| \ 2 /
/ sec |a - --| / | sec |a - --||
\/ \ 2 / \/ \ \ 2 //
$$\sqrt{4 - \frac{8 \sec{\left(a \right)}}{\left(\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(a - \frac{\pi}{2} \right)}}} \sqrt{- \frac{2 \left(- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} + 2}$$
__________________________
/ / 2/ pi\\
/ | cos |a - --||
/ | \ 2 /| _______________________________
/ 2*|1 - ------------| / / pi\
/ | 2 | / 8*cos|a - --|
/ \ cos (a) / / \ 2 /
/ 2 - -------------------- * / 4 - -------------------------
/ 2/ pi\ / / 2/ pi\\
/ cos |a - --| / | cos |a - --||
/ \ 2 / / | \ 2 /|
/ 1 + ------------ / |1 + ------------|*cos(a)
/ 2 / | 2 |
\/ cos (a) \/ \ cos (a) /
$$\sqrt{4 - \frac{8 \cos{\left(a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}\right) \cos{\left(a \right)}}} \sqrt{- \frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}\right)}{1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}} + 2}$$
__________________________
/ / 2/pi \\
/ | csc |-- - a||
/ | \2 /| _______________________________
/ 2*|1 - ------------| / /pi \
/ | 2 | / 8*csc|-- - a|
/ \ csc (a) / / \2 /
/ 2 - -------------------- * / 4 - -------------------------
/ 2/pi \ / / 2/pi \\
/ csc |-- - a| / | csc |-- - a||
/ \2 / / | \2 /|
/ 1 + ------------ / |1 + ------------|*csc(a)
/ 2 / | 2 |
\/ csc (a) \/ \ csc (a) /
$$\sqrt{4 - \frac{8 \csc{\left(- a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right) \csc{\left(a \right)}}} \sqrt{- \frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right)}{1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}} + 2}$$
__________________________________________ |/ 0 for a mod pi = 0|
/ // 0 for 2*a mod pi = 0\ || |
/ || | || /a\ |
/ || 2*cot(a) | || 2*cot|-| |
2* / 4 - 4*|<----------- otherwise | *|< \2/ |
/ || 2 | ||----------- otherwise |
/ ||1 + cot (a) | || 2/a\ |
\/ \\ / ||1 + cot |-| |
|\ \2/ |
$$2 \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 4}\right) \left(\left|{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}}\right|\right)$$
_________________________________________ __________________________________________
/ // 1 for a mod pi = 0\ / // 0 for 2*a mod pi = 0\
/ || | / || |
/ || 2 | / || 2*cot(a) |
/ 2 - 2*|<-1 + cot (a) | * / 4 - 4*|<----------- otherwise |
/ ||------------ otherwise | / || 2 |
/ || 2 | / ||1 + cot (a) |
\/ \\1 + cot (a) / \/ \\ /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
________________________________________ __________________________________________
/ // 1 for a mod pi = 0\ / // 0 for 2*a mod pi = 0\
/ || | / || |
/ || 2 | / || 2*tan(a) |
/ 2 - 2*|<1 - tan (a) | * / 4 - 4*|<----------- otherwise |
/ ||----------- otherwise | / || 2 |
/ || 2 | / ||1 + tan (a) |
\/ \\1 + tan (a) / \/ \\ /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
_________________________________________
/ // 1 for a mod pi = 0\ ___________________________________________________
/ || | / // 0 for 2*a mod pi = 0\
/ || 1 | / || |
/ ||-1 + ------- | / || 2 |
/ || 2 | / ||-------------------- otherwise |
/ 2 - 2*|< tan (a) | * / 4 - 4*|
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(a \right)}}}{1 + \frac{1}{\tan^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \tan{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
______________________________________________________
/ // /pi \ \
/ || 0 for |-- + 2*a| mod pi = 0| __________________________________________
/ || \2 / | / // 0 for 2*a mod pi = 0\
/ || | / || |
/ || / pi\ | / || 2*cot(a) |
/ 2 - 2*|< 2*cot|a + --| | * / 4 - 4*|<----------- otherwise |
/ || \ 4 / | / || 2 |
/ ||---------------- otherwise | / ||1 + cot (a) |
/ || 2/ pi\ | \/ \\ /
/ ||1 + cot |a + --| |
\/ \\ \ 4 / /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
___________________________________________________________
/ // / 3*pi\ \
_________________________________________ / || 1 for |2*a + ----| mod 2*pi = 0|
/ // 1 for a mod pi = 0\ / || \ 2 / |
/ || | / || |
/ || 2 | / || 2/ pi\ |
/ 2 - 2*|<-1 + cot (a) | * / 4 - 4*|<-1 + tan |a + --| |
/ ||------------ otherwise | / || \ 4 / |
/ || 2 | / ||----------------- otherwise |
\/ \\1 + cot (a) / / || 2/ pi\ |
/ || 1 + tan |a + --| |
\/ \\ \ 4 / /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 1 & \text{for}\: \left(2 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
________________________________________________________ ____________________________________________________________
/ // 1 for a mod pi = 0\ / // 0 for 2*a mod pi = 0\
/ || | / || |
/ 2 - 2*|
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
___________________________________________
/ // 1 for a mod pi = 0\
/ || | ______________________________________________________
/ || 2 | / // 0 for 2*a mod pi = 0\
/ || sin (2*a) | / || |
/ ||-1 + --------- | / || sin(2*a) |
/ || 4 | / ||----------------------- otherwise |
/ 2 - 2*|< 4*sin (a) | * / 4 - 4*|
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\sin{\left(2 a \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
____________________________________________________________ _______________________________________________________________
/ // 1 for a mod pi = 0\ / // 0 for 2*a mod pi = 0\
/ || | / || |
/ ||/ 1 for a mod pi = 0 | / ||/ 0 for 2*a mod pi = 0 |
/ ||| | / ||| |
/ 2 - 2*|<| 2 | * / 4 - 4*|<| 2*cot(a) |
/ ||<-1 + cot (a) otherwise | / ||<----------- otherwise otherwise |
/ |||------------ otherwise | / ||| 2 |
/ ||| 2 | / |||1 + cot (a) |
\/ \\\1 + cot (a) / \/ \\\ /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
______________________________________________
/ // 1 for a mod pi = 0\
/ || | _____________________________________________________________
/ || 2 | / // 0 for 2*a mod pi = 0\
/ || cos (a) | / || |
/ ||-1 + ------------ | / || 2*cos(a) |
/ || 2/ pi\ | / ||------------------------------ otherwise |
/ || cos |a - --| | / ||/ 2 \ |
/ 2 - 2*|< \ 2 / | * / 4 - 4*|<| cos (a) | / pi\ |
/ ||----------------- otherwise | / |||1 + ------------|*cos|a - --| |
/ || 2 | / ||| 2/ pi\| \ 2 / |
/ || cos (a) | / ||| cos |a - --|| |
/ || 1 + ------------ | / ||\ \ 2 // |
/ || 2/ pi\ | \/ \\ /
/ || cos |a - --| |
\/ \\ \ 2 / /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cos{\left(a \right)}}{\left(\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
______________________________________________
/ // 1 for a mod pi = 0\ ________________________________________________________
/ || | / // 0 for 2*a mod pi = 0\
/ || 2/ pi\ | / || |
/ || sec |a - --| | / || / pi\ |
/ || \ 2 / | / || 2*sec|a - --| |
/ ||-1 + ------------ | / || \ 2 / |
/ || 2 | / ||------------------------- otherwise |
/ 2 - 2*|< sec (a) | * / 4 - 4*|
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}}{1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \sec{\left(a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
______________________________________________
/ // 1 for a mod pi = 0\
/ || | _____________________________________________________________
/ || 2 | / // 0 for 2*a mod pi = 0\
/ || csc (a) | / || |
/ ||-1 + ------------ | / || 2*csc(a) |
/ || 2/pi \ | / ||------------------------------ otherwise |
/ || csc |-- - a| | / ||/ 2 \ |
/ 2 - 2*|< \2 / | * / 4 - 4*|<| csc (a) | /pi \ |
/ ||----------------- otherwise | / |||1 + ------------|*csc|-- - a| |
/ || 2 | / ||| 2/pi \| \2 / |
/ || csc (a) | / ||| csc |-- - a|| |
/ || 1 + ------------ | / ||\ \2 // |
/ || 2/pi \ | \/ \\ /
/ || csc |-- - a| |
\/ \\ \2 / /
$$\left(\sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 2}\right) \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \csc{\left(a \right)}}{\left(\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
sqrt(2 - 2*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)))*sqrt(4 - 4*Piecewise((0, Mod(2*a = pi, 0)), (2*csc(a)/((1 + csc(a)^2/csc(pi/2 - a)^2)*csc(pi/2 - a)), True)))
Объединение рациональных выражений
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___ ______________ ______________ 2*\/ 2 *\/ 1 - cos(2*a) *\/ 1 - sin(2*a)
$$2 \sqrt{2} \sqrt{- \sin{\left(2 a \right)} + 1} \sqrt{- \cos{\left(2 a \right)} + 1}$$
2*sqrt(2)*sqrt(1 - cos(2*a))*sqrt(1 - sin(2*a))
Численный ответ
[src]
2.82842712474619*(1 - cos(2*a))^0.5*(1 - sin(2*a))^0.5
2.82842712474619*(1 - cos(2*a))^0.5*(1 - sin(2*a))^0.5
Степени
[src]
______________________________ ______________________ / / -2*I*a 2*I*a\ / -2*I*a 2*I*a \/ 4 + 2*I*\- e + e / *\/ 2 - e - e
$$\sqrt{2 i \left(e^{2 i a} - e^{- 2 i a}\right) + 4} \sqrt{- e^{2 i a} + 2 - e^{- 2 i a}}$$
___________________________________ \/ (2 - 2*cos(2*a))*(4 - 4*sin(2*a))
$$\sqrt{\left(- 2 \cos{\left(2 a \right)} + 2\right) \left(- 4 \sin{\left(2 a \right)} + 4\right)}$$
sqrt((2 - 2*cos(2*a))*(4 - 4*sin(2*a)))
Комбинаторика
[src]
___ ______________ ______________ 2*\/ 2 *\/ 1 - cos(2*a) *\/ 1 - sin(2*a)
$$2 \sqrt{2} \sqrt{- \sin{\left(2 a \right)} + 1} \sqrt{- \cos{\left(2 a \right)} + 1}$$
2*sqrt(2)*sqrt(1 - cos(2*a))*sqrt(1 - sin(2*a))
Общий знаменатель
[src]
___ ______________ ______________ 2*\/ 2 *\/ 1 - cos(2*a) *\/ 1 - sin(2*a)
$$2 \sqrt{2} \sqrt{- \sin{\left(2 a \right)} + 1} \sqrt{- \cos{\left(2 a \right)} + 1}$$
2*sqrt(2)*sqrt(1 - cos(2*a))*sqrt(1 - sin(2*a))