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