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Equation:
Equation x/4+13=x+1
Equation 2x^2-x-1=x^2-5x-(-1-x^2)
Equation 2*sin(x)+1=0
Equation x^3-x^2-x-1=0
Express {x} in terms of y where:
8*x+6*y=20
-11*x-19*y=14
-2*x-15*y=-1
5*x+4*y=-12
Identical expressions
sin(pi*(x- three)/ four)= five *pi/ four
sinus of ( Pi multiply by (x minus 3) divide by 4) equally 5 multiply by Pi divide by 4
sinus of ( Pi multiply by (x minus three) divide by four) equally five multiply by Pi divide by four
sin(pi(x-3)/4)=5pi/4
sinpix-3/4=5pi/4
sin(pi*(x-3) divide by 4)=5*pi divide by 4
Similar expressions
tan(pi+x)*cos(pi/3-2x)=tan(5pi/4)
sin(pi*(x+3)/4)=5*pi/4
tg(15pi/4(x)/5)=-(x)/5
cos²(3*x-5pi/4)=1

sin(pi(x-3)/4)=5pi/4 equation

The teacher will be very surprised to see your correct solution 😉

The solution

You have entered [src]

   /pi*(x - 3)\   5*pi
sin|----------| = ----
   \    4     /    4

$$\sin{\left(\frac{\pi \left(x - 3\right)}{4} \right)} = \frac{5 \pi}{4}$$

Simplify

Detail solution

Given the equation
$$\sin{\left(\frac{\pi \left(x - 3\right)}{4} \right)} = \frac{5 \pi}{4}$$
- this is the simplest trigonometric equation

Divide both parts of the equation by -1

The equation is transformed to
$$\sin{\left(\frac{\pi x}{4} + \frac{\pi}{4} \right)} = - \frac{5 \pi}{4}$$
As right part of the equation
modulo =

True

but sin
can no be more than 1 or less than -1
so the solution of the equation d'not exist.

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Rapid solution [src]

                /    /5*pi\\         /    /5*pi\\
       pi + 4*re|asin|----||   4*I*im|asin|----||
                \    \ 4  //         \    \ 4  //
x1 = - --------------------- - ------------------
                 pi                    pi

$$x_{1} = - \frac{\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} - \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}$$

             /    /5*pi\\         /    /5*pi\\
         4*re|asin|----||   4*I*im|asin|----||
             \    \ 4  //         \    \ 4  //
x2 = 3 + ---------------- + ------------------
                pi                  pi

$$x_{2} = \frac{4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + 3 + \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}$$

x2 = 4*re(asin(5*pi/4))/pi + 3 + 4*i*im(asin(5*pi/4))/pi

Sum and product of roots [src]

sum

           /    /5*pi\\         /    /5*pi\\           /    /5*pi\\         /    /5*pi\\
  pi + 4*re|asin|----||   4*I*im|asin|----||       4*re|asin|----||   4*I*im|asin|----||
           \    \ 4  //         \    \ 4  //           \    \ 4  //         \    \ 4  //
- --------------------- - ------------------ + 3 + ---------------- + ------------------
            pi                    pi                      pi                  pi

$$\left(\frac{4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + 3 + \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}\right) + \left(- \frac{\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} - \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}\right)$$

             /    /5*pi\\       /    /5*pi\\
    pi + 4*re|asin|----||   4*re|asin|----||
             \    \ 4  //       \    \ 4  //
3 - --------------------- + ----------------
              pi                   pi

$$- \frac{\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + \frac{4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + 3$$

product

/           /    /5*pi\\         /    /5*pi\\\ /        /    /5*pi\\         /    /5*pi\\\
|  pi + 4*re|asin|----||   4*I*im|asin|----||| |    4*re|asin|----||   4*I*im|asin|----|||
|           \    \ 4  //         \    \ 4  //| |        \    \ 4  //         \    \ 4  //|
|- --------------------- - ------------------|*|3 + ---------------- + ------------------|
\            pi                    pi        / \           pi                  pi        /

$$\left(- \frac{\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} - \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}\right) \left(\frac{4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi} + 3 + \frac{4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}}{\pi}\right)$$

 /         /    /5*pi\\         /    /5*pi\\\ /           /    /5*pi\\         /    /5*pi\\\ 
-|pi + 4*re|asin|----|| + 4*I*im|asin|----|||*|3*pi + 4*re|asin|----|| + 4*I*im|asin|----||| 
 \         \    \ 4  //         \    \ 4  /// \           \    \ 4  //         \    \ 4  /// 
---------------------------------------------------------------------------------------------
                                               2                                             
                                             pi

$$- \frac{\left(\pi + 4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)} + 4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}\right) \left(4 \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)} + 3 \pi + 4 i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{4} \right)}\right)}\right)}{\pi^{2}}$$

-(pi + 4*re(asin(5*pi/4)) + 4*i*im(asin(5*pi/4)))*(3*pi + 4*re(asin(5*pi/4)) + 4*i*im(asin(5*pi/4)))/pi^2

Numerical answer [src]

x1 = -3.0 + 2.60301115305176*i

x2 = 5.0 - 2.60301115305176*i

x2 = 5.0 - 2.60301115305176*i

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Identical expressions

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sin(pi*(x-3)/4)=5*pi/4 equation

Numerical solution:

The solution

sin(pi(x-3)/4)=5pi/4 equation