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