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Equation:
Equation 6x-2x+1=5
Equation x^2+4*x-5=0
Equation x^4-2*x^2-15=0
Equation x^3-3x+2=0
Express {x} in terms of y where:
-11*x+10*y=-3
1*x+6*y=-6
-10*x+13*y=5
8*x+1*y=4
Derivative of:
sin(2*x)
Graphing y =:
sin(2*x)
Integral of d{x}:
sin(2*x)
Identical expressions
one +(sin(x))^ two =sin(two *x)
1 plus ( sinus of (x)) squared equally sinus of (2 multiply by x)
one plus ( sinus of (x)) to the power of two equally sinus of (two multiply by x)
1+(sin(x))2=sin(2*x)
1+sinx2=sin2*x
1+(sin(x))²=sin(2*x)
1+(sin(x)) to the power of 2=sin(2*x)
1+(sin(x))^2=sin(2x)
1+(sin(x))2=sin(2x)
1+sinx2=sin2x
1+sinx^2=sin2x
Similar expressions
2sin^2x-11sinx+9=0
log2((sin2x+2sinx+cosx+1)/(2sinx+1))=0
1-(sin(x))^2=sin(2*x)
1+(sinx)^2=sin(2*x)

1+(sin(x))^2=sin(2*x) equation

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

The solution

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       2              
1 + sin (x) = sin(2*x)

$$\sin^{2}{\left(x \right)} + 1 = \sin{\left(2 x \right)}$$

Simplify

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Sum and product of roots [src]

sum

                                    /   /atan(2)\\                                                                /   /atan(2)\\                         
                                    |sin|-------||               /            /  ___\\                            |sin|-------||                  /  ___\
        /log(5)      /4 ___\\       |   \   2   /|   atan(2)     |log(5)   log\\/ 5 /|              /4 ___\       |   \   2   /|   atan(2)   I*log\\/ 5 /
-pi + I*|------ - log\\/ 5 /| + atan|------------| + ------- + I*|------ - ----------| + -pi - I*log\\/ 5 / + atan|------------| + ------- - ------------
        \  2                /       |   /atan(2)\|      2        \  2          2     /                            |   /atan(2)\|      2           2      
                                    |cos|-------||                                                                |cos|-------||                         
                                    \   \   2   //                                                                \   \   2   //

$$\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} - \frac{i \log{\left(\sqrt{5} \right)}}{2}\right) + \left(\left(- \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}} \right)} - i \log{\left(\sqrt[4]{5} \right)}\right) + \left(\left(- \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{5} \right)} + \frac{\log{\left(5 \right)}}{2}\right)\right) + \left(\frac{\operatorname{atan}{\left(2 \right)}}{2} + i \left(- \frac{\log{\left(\sqrt{5} \right)}}{2} + \frac{\log{\left(5 \right)}}{2}\right)\right)\right)\right)$$

              /   /atan(2)\\                                                                                            
              |sin|-------||                               /            /  ___\\                       /  ___\          
              |   \   2   /|     /log(5)      /4 ___\\     |log(5)   log\\/ 5 /|        /4 ___\   I*log\\/ 5 /          
-2*pi + 2*atan|------------| + I*|------ - log\\/ 5 /| + I*|------ - ----------| - I*log\\/ 5 / - ------------ + atan(2)
              |   /atan(2)\|     \  2                /     \  2          2     /                       2                
              |cos|-------||                                                                                            
              \   \   2   //

$$- 2 \pi + 2 \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}} \right)} + \operatorname{atan}{\left(2 \right)} - i \log{\left(\sqrt[4]{5} \right)} - \frac{i \log{\left(\sqrt{5} \right)}}{2} + i \left(- \log{\left(\sqrt[4]{5} \right)} + \frac{\log{\left(5 \right)}}{2}\right) + i \left(- \frac{\log{\left(\sqrt{5} \right)}}{2} + \frac{\log{\left(5 \right)}}{2}\right)$$

product

/                                    /   /atan(2)\\\                                     /                         /   /atan(2)\\\                         
|                                    |sin|-------||| /            /            /  ___\\\ |                         |sin|-------||| /               /  ___\\
|        /log(5)      /4 ___\\       |   \   2   /|| |atan(2)     |log(5)   log\\/ 5 /|| |           /4 ___\       |   \   2   /|| |atan(2)   I*log\\/ 5 /|
|-pi + I*|------ - log\\/ 5 /| + atan|------------||*|------- + I*|------ - ----------||*|-pi - I*log\\/ 5 / + atan|------------||*|------- - ------------|
|        \  2                /       |   /atan(2)\|| \   2        \  2          2     // |                         |   /atan(2)\|| \   2           2      /
|                                    |cos|-------|||                                     |                         |cos|-------|||                         
\                                    \   \   2   ///                                     \                         \   \   2   ///

$$\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} + i \left(- \frac{\log{\left(\sqrt{5} \right)}}{2} + \frac{\log{\left(5 \right)}}{2}\right)\right) \left(- \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{5} \right)} + \frac{\log{\left(5 \right)}}{2}\right)\right) \left(- \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}} \right)} - i \log{\left(\sqrt[4]{5} \right)}\right) \left(\frac{\operatorname{atan}{\left(2 \right)}}{2} - \frac{i \log{\left(\sqrt{5} \right)}}{2}\right)$$

    4         4             3        2     2        2    2          2       2            2           
atan (2)   log (5)   pi*atan (2)   pi *atan (2)   pi *log (5)   atan (2)*log (5)   pi*log (5)*atan(2)
-------- + ------- - ----------- + ------------ + ----------- + ---------------- - ------------------
   16        256          4             4              16              32                  16

$$- \frac{\pi \operatorname{atan}^{3}{\left(2 \right)}}{4} - \frac{\pi \log{\left(5 \right)}^{2} \operatorname{atan}{\left(2 \right)}}{16} + \frac{\log{\left(5 \right)}^{4}}{256} + \frac{\operatorname{atan}^{4}{\left(2 \right)}}{16} + \frac{\log{\left(5 \right)}^{2} \operatorname{atan}^{2}{\left(2 \right)}}{32} + \frac{\pi^{2} \log{\left(5 \right)}^{2}}{16} + \frac{\pi^{2} \operatorname{atan}^{2}{\left(2 \right)}}{4}$$

atan(2)^4/16 + log(5)^4/256 - pi*atan(2)^3/4 + pi^2*atan(2)^2/4 + pi^2*log(5)^2/16 + atan(2)^2*log(5)^2/32 - pi*log(5)^2*atan(2)/16

Rapid solution [src]

                                         /   /atan(2)\\
                                         |sin|-------||
             /log(5)      /4 ___\\       |   \   2   /|
x1 = -pi + I*|------ - log\\/ 5 /| + atan|------------|
             \  2                /       |   /atan(2)\|
                                         |cos|-------||
                                         \   \   2   //

$$x_{1} = - \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}} \right)} + i \left(- \log{\left(\sqrt[4]{5} \right)} + \frac{\log{\left(5 \right)}}{2}\right)$$

                 /            /  ___\\
     atan(2)     |log(5)   log\\/ 5 /|
x2 = ------- + I*|------ - ----------|
        2        \  2          2     /

$$x_{2} = \frac{\operatorname{atan}{\left(2 \right)}}{2} + i \left(- \frac{\log{\left(\sqrt{5} \right)}}{2} + \frac{\log{\left(5 \right)}}{2}\right)$$

                              /   /atan(2)\\
                              |sin|-------||
                /4 ___\       |   \   2   /|
x3 = -pi - I*log\\/ 5 / + atan|------------|
                              |   /atan(2)\|
                              |cos|-------||
                              \   \   2   //

$$x_{3} = - \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(2 \right)}}{2} \right)}} \right)} - i \log{\left(\sqrt[4]{5} \right)}$$

                    /  ___\
     atan(2)   I*log\\/ 5 /
x4 = ------- - ------------
        2           2

$$x_{4} = \frac{\operatorname{atan}{\left(2 \right)}}{2} - \frac{i \log{\left(\sqrt{5} \right)}}{2}$$

x4 = atan(2)/2 - i*log(sqrt(5))/2

Numerical answer [src]

x1 = -2.58801829469275 + 0.402359478108525*i

x2 = 0.553574358897045 + 0.402359478108525*i

x3 = -2.58801829469275 - 0.402359478108525*i

x4 = 0.553574358897045 - 0.402359478108525*i

x4 = 0.553574358897045 - 0.402359478108525*i

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Identical expressions

Similar expressions

1+(sin(x))^2=sin(2*x) equation

Numerical solution:

The solution