Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- 4*x^2+4*x+1^2 если x=4
- sqrt(4-4*sin(2*a))-sqrt(2-2*cos(2*a)) если a=1/4
- -9*a-16*a если a=1/2
- (y-4)^2-(4-y)*(4+y) если y=-4
- Разложить многочлен на множители:
- 49*a^2*b^4-100*c^4
- 36*d^2-12*d+1
- 35*a*m+27*m*u-15*a*u-63*m^2
- x^2-6*x+8
- Общий знаменатель:
- 4/t-4*z*(t2+16*z2/2-4*tz)
- (m+n-4*m*n/(m+n))/(m/(m+n)-n/(n-m)-2*m*n/m^2-n^2)
- (x-8)/(4*x^2)-(5-12*x)/(6*x^3)
- (z^2/t^2+2*z/t+1)/t+z/t
- Умножение столбиком:
- 600*580
- 60*40
- 60*10
- 24*12
- Деление столбиком:
- 792/6
- 162/8
- 4956/7
- 950/5
- Раскрыть скобки в:
- 2*i*(j+k)+3*j*(i+k)+4*i*(i+j)
- (5*c+2)*(5*c-2)
- (81-69)*(78+23)
- (c+4)*(c-1)-c^2
- Разложение числа на простые множители:
- 28800
- 630
- 6552
- 4752
- Производная:
- 1/4
- Интеграл d{x}:
-
1/4
- График функции y =:
- 1/4
-
Идентичные выражения
- sqrt(четыре - четыре *sin(два *a))-sqrt(два - два *cos(два *a)) если a= один / четыре
- квадратный корень из (4 минус 4 умножить на синус от (2 умножить на a)) минус квадратный корень из (2 минус 2 умножить на косинус от (2 умножить на a)) если a равно 1 делить на 4
- квадратный корень из (четыре минус четыре умножить на синус от (два умножить на a)) минус квадратный корень из (два минус два умножить на косинус от (два умножить на a)) если a равно один делить на четыре
- √(4-4*sin(2*a))-√(2-2*cos(2*a)) если a=1/4
- sqrt(4-4sin(2a))-sqrt(2-2cos(2a)) если a=1/4
- sqrt4-4sin2a-sqrt2-2cos2a если a=1/4
- sqrt(4-4*sin(2*a))-sqrt(2-2*cos(2*a)) при a=1/4
- sqrt(4-4*sin(2*a))-sqrt(2-2*cos(2*a)) если a=1 разделить на 4
-
Похожие выражения
- sqrt(4+4*sin(2*a))-sqrt(2-2*cos(2*a)) если a=1/4
- sqrt(4-4*sin(2*a))-sqrt(2+2*cos(2*a)) если a=1/4
- sqrt(4-4*sin(2*a))+sqrt(2-2*cos(2*a)) если a=1/4
sqrt(4-4*sin(2*a))-sqrt(2-2*cos(2*a)) если a=1/4
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 ______________
- 2*\/ sin (a) + 2*\/ 1 - sin(2*a)
$$2 \sqrt{- \sin{\left(2 a \right)} + 1} - 2 \sqrt{\sin^{2}{\left(a \right)}}$$
-2*sqrt(sin(a)^2) + 2*sqrt(1 - sin(2*a))
Подстановка условия
[src]
sqrt(4 - 4*sin(2*a)) - sqrt(2 - 2*cos(2*a)) при a = 1/4
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 ______________
- 2*\/ sin (a) + 2*\/ 1 - sin(2*a)
$$2 \sqrt{- \sin{\left(2 a \right)} + 1} - 2 \sqrt{\sin^{2}{\left(a \right)}}$$
переменные
a = 1/4
$$a = \frac{1}{4}$$
_____________
/ 2 __________________
- 2*\/ sin ((1/4)) + 2*\/ 1 - sin(2*(1/4))
$$2 \sqrt{- \sin{\left(2 (1/4) \right)} + 1} - 2 \sqrt{\sin^{2}{\left((1/4) \right)}}$$
___________
/ 2 ________________
- 2*\/ sin (1/4) + 2*\/ 1 - sin(2*1/4)
$$- 2 \sqrt{\sin^{2}{\left(\frac{1}{4} \right)}} + 2 \sqrt{- \sin{\left(2 \cdot \frac{1}{4} \right)} + 1}$$
______________ -2*sin(1/4) + 2*\/ 1 - sin(1/2)
$$- 2 \sin{\left(\frac{1}{4} \right)} + 2 \sqrt{- \sin{\left(\frac{1}{2} \right)} + 1}$$
-2*sin(1/4) + 2*sqrt(1 - sin(1/2))
Общий знаменатель
[src]
______________ ___ ______________ 2*\/ 1 - sin(2*a) - \/ 2 *\/ 1 - cos(2*a)
$$2 \sqrt{- \sin{\left(2 a \right)} + 1} - \sqrt{2} \sqrt{- \cos{\left(2 a \right)} + 1}$$
2*sqrt(1 - sin(2*a)) - sqrt(2)*sqrt(1 - cos(2*a))
Объединение рациональных выражений
[src]
______________ ___ ______________ 2*\/ 1 - sin(2*a) - \/ 2 *\/ 1 - cos(2*a)
$$2 \sqrt{- \sin{\left(2 a \right)} + 1} - \sqrt{2} \sqrt{- \cos{\left(2 a \right)} + 1}$$
2*sqrt(1 - sin(2*a)) - sqrt(2)*sqrt(1 - cos(2*a))
Комбинаторика
[src]
______________ ___ ______________ 2*\/ 1 - sin(2*a) - \/ 2 *\/ 1 - cos(2*a)
$$2 \sqrt{- \sin{\left(2 a \right)} + 1} - \sqrt{2} \sqrt{- \cos{\left(2 a \right)} + 1}$$
2*sqrt(1 - sin(2*a)) - sqrt(2)*sqrt(1 - cos(2*a))
Степени
[src]
______________________________ ______________________ / / -2*I*a 2*I*a\ / -2*I*a 2*I*a \/ 4 + 2*I*\- e + e / - \/ 2 - e - e
$$\sqrt{2 i \left(e^{2 i a} - e^{- 2 i a}\right) + 4} - \sqrt{- e^{2 i a} + 2 - e^{- 2 i a}}$$
sqrt(4 + 2*i*(-exp(-2*i*a) + exp(2*i*a))) - sqrt(2 - exp(-2*i*a) - exp(2*i*a))
Тригонометрическая часть
[src]
______________ -2*|sin(a)| + 2*\/ 1 - sin(2*a)
$$2 \sqrt{- \sin{\left(2 a \right)} + 1} - 2 \left|{\sin{\left(a \right)}}\right|$$
______________ ______________ / 4 / 2 / 4 - -------- - / 2 - -------- \/ csc(2*a) \/ sec(2*a)
$$- \sqrt{2 - \frac{2}{\sec{\left(2 a \right)}}} + \sqrt{4 - \frac{4}{\csc{\left(2 a \right)}}}$$
______________
2 / 1
- -------- + 2* / 1 - --------
|csc(a)| \/ csc(2*a)
$$2 \sqrt{1 - \frac{1}{\csc{\left(2 a \right)}}} - \frac{2}{\left|{\csc{\left(a \right)}}\right|}$$
_____________________ / / pi\ ________________ / 4 - 4*cos|2*a - --| - \/ 2 - 2*cos(2*a) \/ \ 2 /
$$- \sqrt{- 2 \cos{\left(2 a \right)} + 2} + \sqrt{- 4 \cos{\left(2 a - \frac{\pi}{2} \right)} + 4}$$
_____________________
________________ / /pi \
\/ 4 - 4*sin(2*a) - / 2 - 2*sin|-- + 2*a|
\/ \2 /
$$- \sqrt{- 2 \sin{\left(2 a + \frac{\pi}{2} \right)} + 2} + \sqrt{- 4 \sin{\left(2 a \right)} + 4}$$
___________________ ______________
/ 4 / 2
/ 4 - ------------- - / 2 - --------
/ / pi\ \/ sec(2*a)
/ 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*|cos|a - --|| + 2* / 1 - cos|2*a - --|
| \ 2 /| \/ \ 2 /
$$2 \sqrt{- \cos{\left(2 a - \frac{\pi}{2} \right)} + 1} - 2 \left|{\cos{\left(a - \frac{\pi}{2} \right)}}\right|$$
___________________ ______________
/ 4 / 2
/ 4 - ------------- - / 2 - --------
/ /pi \ \/ sec(2*a)
/ sec|-- - 2*a|
\/ \2 /
$$- \sqrt{2 - \frac{2}{\sec{\left(2 a \right)}}} + \sqrt{4 - \frac{4}{\sec{\left(- 2 a + \frac{\pi}{2} \right)}}}$$
______________ ___________________
/ 4 / 2
/ 4 - -------- - / 2 - -------------
\/ csc(2*a) / /pi \
/ csc|-- - 2*a|
\/ \2 /
$$- \sqrt{2 - \frac{2}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}} + \sqrt{4 - \frac{4}{\csc{\left(2 a \right)}}}$$
___________________ ___________________
/ 4 / 2
/ 4 - ------------- - / 2 - -------------
\/ csc(pi - 2*a) / /pi \
/ csc|-- - 2*a|
\/ \2 /
$$- \sqrt{2 - \frac{2}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}} + \sqrt{4 - \frac{4}{\csc{\left(- 2 a + \pi \right)}}}$$
______________ 2/a\ | /a\|
2*\/ 1 - sin(2*a) - 4*cos |-|*|tan|-||
\2/ | \2/|
$$- 4 \cos^{2}{\left(\frac{a}{2} \right)} \left|{\tan{\left(\frac{a}{2} \right)}}\right| + 2 \sqrt{- \sin{\left(2 a \right)} + 1}$$
___________________
2 / 1
- ------------- + 2* / 1 - -------------
| / pi\| / / pi\
|sec|a - --|| / sec|2*a - --|
| \ 2 /| \/ \ 2 /
$$2 \sqrt{1 - \frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}}} - \frac{2}{\left|{\sec{\left(a - \frac{\pi}{2} \right)}}\right|}$$
___________________________ _____________________ / 2 2 \/ 4 - 8*cos(a)*sin(a) - \/ 2 - 2*cos (a) + 2*sin (a)
$$\sqrt{- 8 \sin{\left(a \right)} \cos{\left(a \right)} + 4} - \sqrt{2 \sin^{2}{\left(a \right)} - 2 \cos^{2}{\left(a \right)} + 2}$$
| /a\|
_________________ 4*|tan|-||
/ 2*tan(a) | \2/|
2* / 1 - ----------- - -----------
/ 2 2/a\
\/ 1 + tan (a) 1 + tan |-|
\2/
$$2 \sqrt{1 - \frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}} - \frac{4 \left|{\tan{\left(\frac{a}{2} \right)}}\right|}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
_____________________
_________________ / / 2 \
/ 8*tan(a) / 2*\1 - tan (a)/
/ 4 - ----------- - / 2 - ---------------
/ 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 + --|
/ 8*cot(a) / \ 4 /
/ 4 - ----------- - / 2 - ----------------
/ 2 / 2/ pi\
\/ 1 + cot (a) / 1 + tan |a + --|
\/ \ 4 /
$$- \sqrt{2 - \frac{4 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}} + \sqrt{4 - \frac{8 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}}$$
______________________
/ / pi\
_________________ / 4*tan|a + --|
/ 8*tan(a) / \ 4 /
/ 4 - ----------- - / 2 - ----------------
/ 2 / 2/ pi\
\/ 1 + tan (a) / 1 + tan |a + --|
\/ \ 4 /
$$- \sqrt{2 - \frac{4 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}} + \sqrt{4 - \frac{8 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}}$$
_____________________
/ / 1 \
/ 2*|1 - -------|
__________________________ / | 2 |
/ 8 / \ cot (a)/
/ 4 - -------------------- - / 2 - ---------------
/ / 1 \ / 1
/ |1 + -------|*cot(a) / 1 + -------
/ | 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\\ ______________________
/ 4*|-1 + tan |a + --|| / / 2 \
/ \ \ 4 // / 2*\-1 + cot (a)/
/ 4 - --------------------- - / 2 - ----------------
/ 2/ pi\ / 2
/ 1 + tan |a + --| \/ 1 + cot (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\\ _____________________
/ 4*|1 - cot |a + --|| / / 2 \
/ \ \ 4 // / 2*\1 - tan (a)/
/ 4 - -------------------- - / 2 - ---------------
/ 2/ pi\ / 2
/ 1 + cot |a + --| \/ 1 + tan (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 a mod pi = 0| / // 0 for 2*a mod pi = 0\
- 2*|< | + 2* / 1 - |< |
|\sin(a) otherwise | \/ \\sin(2*a) otherwise /
$$\left(2 \left(\sqrt{\left(- \begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1}\right)\right) - \left(2 \left(\left|{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}\right|\right)\right)$$
_______________________________________ _____________________________________ / // 0 for 2*a mod pi = 0\ / // 1 for a mod pi = 0\ / 4 - 4*|< | - / 2 - 2*|< | \/ \\sin(2*a) otherwise / \/ \\cos(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 |
/ 16*sin (a) / \ sin (2*a)/
/ 4 - ------------------------ - / 2 - -----------------
/ / 4 \ / 4
/ | 4*sin (a)| / 4*sin (a)
/ |1 + ---------|*sin(2*a) / 1 + ---------
/ | 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\ / || |
/ 4 - 4*|< | - / 2 - 2*|< /pi \ |
\/ \\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\
/ 4 - 4*|< / pi\ | - / 2 - 2*|< |
/ ||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|
/ 4 - 4*|< | - / 2 - 2*|< \2 / |
\/ \\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 |2*a + ----| mod 2*pi = 0| / // 1 for a mod pi = 0\
/ 4 - 4*|< \ 2 / | - / 2 - 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 | / || |
/ 4 - 4*|<------------- otherwise | - / 2 - 2*|< 1 |
/ || / pi\ | / ||-------- otherwise |
/ ||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 |
/ 4 - 4*|< 1 | - / 2 - 2*|<------------- otherwise |
/ ||-------- otherwise | / || /pi \ |
\/ \\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 - --||
/ 8*sec(a) / \ \ 2 //
/ 4 - ------------------------------ - / 2 - --------------------
/ / 2 \ / 2
/ | sec (a) | / pi\ / sec (a)
/ |1 + ------------|*sec|a - --| / 1 + ------------
/ | 2/ pi\| \ 2 / / 2/ pi\
/ | 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 /|
/ / pi\ / 2*|1 - ------------|
/ 8*cos|a - --| / | 2 |
/ \ 2 / / \ cos (a) /
/ 4 - ------------------------- - / 2 - --------------------
/ / 2/ pi\\ / 2/ pi\
/ | cos |a - --|| / cos |a - --|
/ | \ 2 /| / \ 2 /
/ |1 + ------------|*cos(a) / 1 + ------------
/ | 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 /|
/ /pi \ / 2*|1 - ------------|
/ 8*csc|-- - a| / | 2 |
/ \2 / / \ csc (a) /
/ 4 - ------------------------- - / 2 - --------------------
/ / 2/pi \\ / 2/pi \
/ | csc |-- - a|| / csc |-- - a|
/ | \2 /| / \2 /
/ |1 + ------------|*csc(a) / 1 + ------------
/ | 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 2*a mod pi = 0\ / // 1 for a mod pi = 0\
/ || | / || |
/ || 2*cot(a) | / || 2 |
/ 4 - 4*|<----------- otherwise | - / 2 - 2*|<-1 + cot (a) |
/ || 2 | / ||------------ otherwise |
/ ||1 + cot (a) | / || 2 |
\/ \\ / \/ \\1 + cot (a) /
$$\left(- \sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 2}\right) + \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
|/ 0 for a mod pi = 0| ________________________________________
|| | / // 0 for 2*a mod pi = 0\
|| /a\ | / || |
|| 2*cot|-| | / || 2*cot(a) |
- 2*|< \2/ | + 2* / 1 - |<----------- otherwise |
||----------- otherwise | / || 2 |
|| 2/a\ | / ||1 + cot (a) |
||1 + cot |-| | \/ \\ /
|\ \2/ |
$$\left(2 \left(\sqrt{\left(- \begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)\right) - \left(2 \left(\left|{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}}\right|\right)\right)$$
__________________________________________ ________________________________________
/ // 0 for 2*a mod pi = 0\ / // 1 for a mod pi = 0\
/ || | / || |
/ || 2*tan(a) | / || 2 |
/ 4 - 4*|<----------- otherwise | - / 2 - 2*|<1 - tan (a) |
/ || 2 | / ||----------- otherwise |
/ ||1 + tan (a) | / || 2 |
\/ \\ / \/ \\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 |
/ || 2 | / ||-1 + ------- |
/ ||-------------------- otherwise | / || 2 |
/ 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|
/ // 0 for 2*a mod pi = 0\ / || \2 / |
/ || | / || |
/ || 2*cot(a) | / || / pi\ |
/ 4 - 4*|<----------- otherwise | - / 2 - 2*|< 2*cot|a + --| |
/ || 2 | / || \ 4 / |
/ ||1 + cot (a) | / ||---------------- otherwise |
\/ \\ / / || 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| _________________________________________
/ || \ 2 / | / // 1 for a mod pi = 0\
/ || | / || |
/ || 2/ pi\ | / || 2 |
/ 4 - 4*|<-1 + tan |a + --| | - / 2 - 2*|<-1 + cot (a) |
/ || \ 4 / | / ||------------ otherwise |
/ ||----------------- otherwise | / || 2 |
/ || 2/ pi\ | \/ \\1 + cot (a) /
/ || 1 + tan |a + --| |
\/ \\ \ 4 / /
$$\left(- \sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 2}\right) + \left(\sqrt{\left(- 4 \left(\begin{cases} 1 & \text{for}\: \left(2 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
____________________________________________________________ ________________________________________________________
/ // 0 for 2*a mod pi = 0\ / // 1 for a mod pi = 0\
/ || | / || |
/ 4 - 4*|
$$\left(- \sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 2}\right) + \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
___________________________________________
/ // 1 for a mod pi = 0\
______________________________________________________ / || |
/ // 0 for 2*a mod pi = 0\ / || 2 |
/ || | / || sin (2*a) |
/ || sin(2*a) | / ||-1 + --------- |
/ ||----------------------- otherwise | / || 4 |
/ 4 - 4*|
$$\left(- \sqrt{\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)\right) + 2}\right) + \left(\sqrt{\left(- 4 \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\sin{\left(2 a \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 4}\right)$$
_______________________________________________________________ ____________________________________________________________
/ // 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 |
/ ||| | / ||| |
/ 4 - 4*|<| 2*cot(a) | - / 2 - 2*|<| 2 |
/ ||<----------- otherwise otherwise | / ||<-1 + cot (a) otherwise |
/ ||| 2 | / |||------------ otherwise |
/ |||1 + cot (a) | / ||| 2 |
\/ \\\ / \/ \\\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\
_____________________________________________________________ / || |
/ // 0 for 2*a mod pi = 0\ / || 2 |
/ || | / || cos (a) |
/ || 2*cos(a) | / ||-1 + ------------ |
/ ||------------------------------ otherwise | / || 2/ pi\ |
/ ||/ 2 \ | / || cos |a - --| |
/ 4 - 4*|<| cos (a) | / pi\ | - / 2 - 2*|< \ 2 / |
/ |||1 + ------------|*cos|a - --| | / ||----------------- otherwise |
/ ||| 2/ pi\| \ 2 / | / || 2 |
/ ||| cos |a - --|| | / || cos (a) |
/ ||\ \ 2 // | / || 1 + ------------ |
\/ \\ / / || 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\ |
/ || / pi\ | / || sec |a - --| |
/ || 2*sec|a - --| | / || \ 2 / |
/ || \ 2 / | / ||-1 + ------------ |
/ ||------------------------- otherwise | / || 2 |
/ 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\
_____________________________________________________________ / || |
/ // 0 for 2*a mod pi = 0\ / || 2 |
/ || | / || csc (a) |
/ || 2*csc(a) | / ||-1 + ------------ |
/ ||------------------------------ otherwise | / || 2/pi \ |
/ ||/ 2 \ | / || csc |-- - a| |
/ 4 - 4*|<| csc (a) | /pi \ | - / 2 - 2*|< \2 / |
/ |||1 + ------------|*csc|-- - a| | / ||----------------- otherwise |
/ ||| 2/pi \| \2 / | / || 2 |
/ ||| csc |-- - a|| | / || csc (a) |
/ ||\ \2 // | / || 1 + ------------ |
\/ \\ / / || 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(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))) - 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)))
Численный ответ
[src]
2.0*(1 - sin(2*a))^0.5 - 1.4142135623731*(1 - cos(2*a))^0.5
2.0*(1 - sin(2*a))^0.5 - 1.4142135623731*(1 - cos(2*a))^0.5