Господин Экзамен
cos(90+t) если t=2
Решение
Подстановка условия
[src]
cos(90 + t) при t = 2
подставляем
cos(90 + t)
$$\cos{\left(t + 90 \right)}$$
cos(90 + t)
$$\cos{\left(t + 90 \right)}$$
переменные
t = 2
$$t = 2$$
cos(90 + (2))
$$\cos{\left((2) + 90 \right)}$$
cos(90 + 2)
$$\cos{\left(2 + 90 \right)}$$
cos(92)
$$\cos{\left(92 \right)}$$
cos(92)
Тригонометрическая часть
[src]
1 ----------- sec(90 + t)
$$\frac{1}{\sec{\left(t + 90 \right)}}$$
/ pi\ sin|90 + t + --| \ 2 /
$$\sin{\left(t + \frac{\pi}{2} + 90 \right)}$$
1 ----------------- / pi \ csc|-90 + -- - t| \ 2 /
$$\frac{1}{\csc{\left(- t - 90 + \frac{\pi}{2} \right)}}$$
2/ t\
-1 + cot |45 + -|
\ 2/
-----------------
2/ t\
1 + cot |45 + -|
\ 2/
$$\frac{\cot^{2}{\left(\frac{t}{2} + 45 \right)} - 1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)} + 1}$$
2/ t\
1 - tan |45 + -|
\ 2/
----------------
2/ t\
1 + tan |45 + -|
\ 2/
$$\frac{- \tan^{2}{\left(\frac{t}{2} + 45 \right)} + 1}{\tan^{2}{\left(\frac{t}{2} + 45 \right)} + 1}$$
1
1 - ------------
2/ t\
cot |45 + -|
\ 2/
----------------
1
1 + ------------
2/ t\
cot |45 + -|
\ 2/
$$\frac{1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)}}}$$
/ t pi\
2*tan|45 + - + --|
\ 2 4 /
---------------------
2/ t pi\
1 + tan |45 + - + --|
\ 2 4 /
$$\frac{2 \tan{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)}}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)} + 1}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 < \cos(90 + t) otherwise
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\cos{\left(t + 90 \right)} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | < 1 |----------- otherwise \sec(90 + t)
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(t + 90 \right)}} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | < / pi\ |sin|90 + t + --| otherwise \ \ 2 /
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\sin{\left(t + \frac{\pi}{2} + 90 \right)} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | | 1 <----------------- otherwise | / pi \ |csc|-90 + -- - t| \ \ 2 /
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- t - 90 + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$
4/ t\
4*sin |45 + -|
\ 2/
1 - --------------
2
sin (90 + t)
------------------
4/ t\
4*sin |45 + -|
\ 2/
1 + --------------
2
sin (90 + t)
$$\frac{- \frac{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)}}{\sin^{2}{\left(t + 90 \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)}}{\sin^{2}{\left(t + 90 \right)}} + 1}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | | 2/ t\ |-1 + cot |45 + -| < \ 2/ |----------------- otherwise | 2/ t\ | 1 + cot |45 + -| \ \ 2/
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} + 45 \right)} - 1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)} + 1} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | | 2/ t\ |1 - tan |45 + -| < \ 2/ |---------------- otherwise | 2/ t\ |1 + tan |45 + -| \ \ 2/
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{t}{2} + 45 \right)} + 1}{\tan^{2}{\left(\frac{t}{2} + 45 \right)} + 1} & \text{otherwise} \end{cases}$$
/ / 55*pi\ | 0 for |90 + t - -----| mod pi = 0 | \ 2 / < | / t pi\ |(1 + sin(90 + t))*cot|45 + - + --| otherwise \ \ 2 4 /
$$\begin{cases} 0 & \text{for}\: \left(t - \frac{55 \pi}{2} + 90\right) \bmod \pi = 0 \\\left(\sin{\left(t + 90 \right)} + 1\right) \cot{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)} & \text{otherwise} \end{cases}$$
2/ t pi\
cos |45 + - - --|
\ 2 2 /
1 - -----------------
2/ t\
cos |45 + -|
\ 2/
---------------------
2/ t pi\
cos |45 + - - --|
\ 2 2 /
1 + -----------------
2/ t\
cos |45 + -|
\ 2/
$$\frac{1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}}{\cos^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}}{\cos^{2}{\left(\frac{t}{2} + 45 \right)}}}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | | 1 |-1 + ------------ | 2/ t\ | tan |45 + -| < \ 2/ |----------------- otherwise | 1 | 1 + ------------ | 2/ t\ | tan |45 + -| \ \ 2/
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} + 45 \right)}}} & \text{otherwise} \end{cases}$$
2/ t\
sec |45 + -|
\ 2/
1 - -----------------
2/ t pi\
sec |45 + - - --|
\ 2 2 /
---------------------
2/ t\
sec |45 + -|
\ 2/
1 + -----------------
2/ t pi\
sec |45 + - - --|
\ 2 2 /
$$\frac{- \frac{\sec^{2}{\left(\frac{t}{2} + 45 \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}} + 1}{\frac{\sec^{2}{\left(\frac{t}{2} + 45 \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}} + 1}$$
2/ pi t\
csc |-45 + -- - -|
\ 2 2/
1 - ------------------
2/ t\
csc |45 + -|
\ 2/
----------------------
2/ pi t\
csc |-45 + -- - -|
\ 2 2/
1 + ------------------
2/ t\
csc |45 + -|
\ 2/
$$\frac{1 - \frac{\csc^{2}{\left(- \frac{t}{2} - 45 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{t}{2} - 45 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} + 45 \right)}}}$$
/ / 55*pi\ | 0 for |90 + t - -----| mod pi = 0 | \ 2 / | | / t pi\ < 2*cot|45 + - + --| | \ 2 4 / |--------------------- otherwise | 2/ t pi\ |1 + cot |45 + - + --| \ \ 2 4 /
$$\begin{cases} 0 & \text{for}\: \left(t - \frac{55 \pi}{2} + 90\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)}}{\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)} + 1} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0
|
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\cos{\left(t + 90 \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | | 2 4/ t\ |sin (90 + t) - 4*sin |45 + -| < \ 2/ |----------------------------- otherwise | 2 4/ t\ |sin (90 + t) + 4*sin |45 + -| \ \ 2/
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{- 4 \sin^{4}{\left(\frac{t}{2} + 45 \right)} + \sin^{2}{\left(t + 90 \right)}}{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)} + \sin^{2}{\left(t + 90 \right)}} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | | 2 | sin (90 + t) |-1 + -------------- | 4/ t\ | 4*sin |45 + -| < \ 2/ |------------------- otherwise | 2 | sin (90 + t) | 1 + -------------- | 4/ t\ | 4*sin |45 + -| \ \ 2/
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(t + 90 \right)}}{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{\sin^{2}{\left(t + 90 \right)}}{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)}}} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | |/ 1 for (90 + t - 28*pi) mod 2*pi = 0 || || 2/ t\ <|-1 + cot |45 + -| |< \ 2/ otherwise ||----------------- otherwise || 2/ t\ || 1 + cot |45 + -| \\ \ 2/
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} + 45 \right)} - 1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | | 2/ t\ | cos |45 + -| | \ 2/ |-1 + ----------------- | 2/ t pi\ | cos |45 + - - --| < \ 2 2 / |---------------------- otherwise | 2/ t\ | cos |45 + -| | \ 2/ |1 + ----------------- | 2/ t pi\ | cos |45 + - - --| \ \ 2 2 /
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{t}{2} + 45 \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}} - 1}{\frac{\cos^{2}{\left(\frac{t}{2} + 45 \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}} + 1} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | | 2/ t pi\ | sec |45 + - - --| | \ 2 2 / |-1 + ----------------- | 2/ t\ | sec |45 + -| < \ 2/ |---------------------- otherwise | 2/ t pi\ | sec |45 + - - --| | \ 2 2 / |1 + ----------------- | 2/ t\ | sec |45 + -| \ \ 2/
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}}{\sec^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}}{\sec^{2}{\left(\frac{t}{2} + 45 \right)}}} & \text{otherwise} \end{cases}$$
/ 1 for (90 + t - 28*pi) mod 2*pi = 0 | | 2/ t\ | csc |45 + -| | \ 2/ |-1 + ------------------ | 2/ pi t\ | csc |-45 + -- - -| < \ 2 2/ |----------------------- otherwise | 2/ t\ | csc |45 + -| | \ 2/ | 1 + ------------------ | 2/ pi t\ | csc |-45 + -- - -| \ \ 2 2/
$$\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{t}{2} + 45 \right)}}{\csc^{2}{\left(- \frac{t}{2} - 45 + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{t}{2} + 45 \right)}}{\csc^{2}{\left(- \frac{t}{2} - 45 + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}$$
Piecewise((1, Mod(90 + t - 28*pi = 2*pi, 0)), ((-1 + csc(45 + t/2)^2/csc(-45 + pi/2 - t/2)^2)/(1 + csc(45 + t/2)^2/csc(-45 + pi/2 - t/2)^2), True))
Степени
[src]
I*(-90 - t) I*(90 + t)
e e
------------ + -----------
2 2
$$\frac{e^{i \left(- t - 90\right)}}{2} + \frac{e^{i \left(t + 90\right)}}{2}$$
exp(i*(-90 - t))/2 + exp(i*(90 + t))/2