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La ecuación:
Ecuación x^2-4x+5=0
Ecuación x^2+3x+2=0
Ecuación x^2+3x=4
Ecuación -x^2+5*x-6=0
Expresar {x} en función de y en la ecuación:
10*x-2*y=0
1*x+4*y=20
3*x-7*y=13
13*x-15*y=-6
Expresiones idénticas
sec^ dos y/cos^ dos (2x)=2
sec al cuadrado y dividir por coseno de al cuadrado (2x) es igual a 2
sec en el grado dos y dividir por coseno de en el grado dos (2x) es igual a 2
sec2y/cos2(2x)=2
sec2y/cos22x=2
sec²y/cos²(2x)=2
sec en el grado 2y/cos en el grado 2(2x)=2
sec^2y/cos^22x=2
sec^2y dividir por cos^2(2x)=2
Expresiones con funciones
sec
sec(x)=5/2
Coseno cos
cos(x)=(-1)/sqrt(2)
cos(2*x)+3*sin(x)-2=0
cos(-x)=-1
cos(2x)=1/2
cos(4*x)=1

sec^2y/cos^2(2x)=2 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

    2        
 sec (y)     
--------- = 2
   2         
cos (2*x)

$$\frac{\sec^{2}{\left(y \right)}}{\cos^{2}{\left(2 x \right)}} = 2$$

Simplificar

Solución detallada

Tenemos la ecuación
$$\frac{\sec^{2}{\left(y \right)}}{\cos^{2}{\left(2 x \right)}} = 2$$
cambiamos
$$-2 + \frac{\sec^{2}{\left(y \right)}}{\cos^{2}{\left(2 x \right)}} = 0$$
$$-2 + \frac{\sec^{2}{\left(y \right)}}{\cos^{2}{\left(2 x \right)}} = 0$$
Sustituimos
$$w = \sec{\left(y \right)}$$
Es la ecuación de la forma

a*w^2 + b*w + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = \frac{1}{\cos^{2}{\left(2 x \right)}}$$
$$b = 0$$
$$c = -2$$
, entonces

D = b^2 - 4 * a * c =

(0)^2 - 4 * (cos(2*x)^(-2)) * (-2) = 8/cos(2*x)^2

La ecuación tiene dos raíces.

w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

o
$$w_{1} = \sqrt{2} \sqrt{\frac{1}{\cos^{2}{\left(2 x \right)}}} \cos^{2}{\left(2 x \right)}$$
$$w_{2} = - \sqrt{2} \sqrt{\frac{1}{\cos^{2}{\left(2 x \right)}}} \cos^{2}{\left(2 x \right)}$$
hacemos cambio inverso
$$\sec{\left(y \right)} = w$$
sustituimos w:

Gráfica

Respuesta rápida [src]

            /|                      _________________|\      /       _________________\
            ||    ___        ___   /          2      ||      |      /          2      |
            ||  \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |1 - \/  1 - 2*cos (2*x) |
y1 = - I*log||---------- - --------------------------|| + arg|------------------------|
            \|2*cos(2*x)           2*cos(2*x)        |/      \        cos(2*x)        /

$$y_{1} = - i \log{\left(\left|{- \frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} + \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{1 - \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{\cos{\left(2 x \right)}} \right)}$$

            /|                        _________________|\      /        _________________\
            ||      ___        ___   /          2      ||      |       /          2      |
            ||    \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |-1 + \/  1 - 2*cos (2*x) |
y2 = - I*log||- ---------- + --------------------------|| + arg|-------------------------|
            \|  2*cos(2*x)           2*cos(2*x)        |/      \         cos(2*x)        /

$$y_{2} = - i \log{\left(\left|{\frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} - \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} - 1}{\cos{\left(2 x \right)}} \right)}$$

            /|                        _________________|\      / /       _________________\ \
            ||      ___        ___   /          2      ||      | |      /          2      | |
            ||    \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |-\1 + \/  1 - 2*cos (2*x) / |
y3 = - I*log||- ---------- - --------------------------|| + arg|----------------------------|
            \|  2*cos(2*x)           2*cos(2*x)        |/      \          cos(2*x)          /

$$y_{3} = - i \log{\left(\left|{- \frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} - \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(- \frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)}$$

            /|                      _________________|\      /       _________________\
            ||    ___        ___   /          2      ||      |      /          2      |
            ||  \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |1 + \/  1 - 2*cos (2*x) |
y4 = - I*log||---------- + --------------------------|| + arg|------------------------|
            \|2*cos(2*x)           2*cos(2*x)        |/      \        cos(2*x)        /

$$y_{4} = - i \log{\left(\left|{\frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} + \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)}$$

y4 = -i*log(Abs(sqrt(2)*sqrt(1 - 2*cos(2*x)^2)/(2*cos(2*x)) + sqrt(2)/(2*cos(2*x)))) + arg((sqrt(1 - 2*cos(2*x)^2) + 1)/cos(2*x))

Suma y producto de raíces [src]

suma

       /|                      _________________|\      /       _________________\          /|                        _________________|\      /        _________________\          /|                        _________________|\      / /       _________________\ \          /|                      _________________|\      /       _________________\
       ||    ___        ___   /          2      ||      |      /          2      |          ||      ___        ___   /          2      ||      |       /          2      |          ||      ___        ___   /          2      ||      | |      /          2      | |          ||    ___        ___   /          2      ||      |      /          2      |
       ||  \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |1 - \/  1 - 2*cos (2*x) |          ||    \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |-1 + \/  1 - 2*cos (2*x) |          ||    \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |-\1 + \/  1 - 2*cos (2*x) / |          ||  \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |1 + \/  1 - 2*cos (2*x) |
- I*log||---------- - --------------------------|| + arg|------------------------| + - I*log||- ---------- + --------------------------|| + arg|-------------------------| + - I*log||- ---------- - --------------------------|| + arg|----------------------------| + - I*log||---------- + --------------------------|| + arg|------------------------|
       \|2*cos(2*x)           2*cos(2*x)        |/      \        cos(2*x)        /          \|  2*cos(2*x)           2*cos(2*x)        |/      \         cos(2*x)        /          \|  2*cos(2*x)           2*cos(2*x)        |/      \          cos(2*x)          /          \|2*cos(2*x)           2*cos(2*x)        |/      \        cos(2*x)        /

$$\left(- i \log{\left(\left|{\frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} + \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)}\right) + \left(\left(- i \log{\left(\left|{- \frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} - \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(- \frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)}\right) + \left(\left(- i \log{\left(\left|{- \frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} + \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{1 - \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{\cos{\left(2 x \right)}} \right)}\right) + \left(- i \log{\left(\left|{\frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} - \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} - 1}{\cos{\left(2 x \right)}} \right)}\right)\right)\right)$$

       /|                      _________________|\        /|                      _________________|\        /|                        _________________|\        /|                        _________________|\      /       _________________\      /       _________________\      /        _________________\      / /       _________________\ \
       ||    ___        ___   /          2      ||        ||    ___        ___   /          2      ||        ||      ___        ___   /          2      ||        ||      ___        ___   /          2      ||      |      /          2      |      |      /          2      |      |       /          2      |      | |      /          2      | |
       ||  \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||        ||  \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||        ||    \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||        ||    \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |1 + \/  1 - 2*cos (2*x) |      |1 - \/  1 - 2*cos (2*x) |      |-1 + \/  1 - 2*cos (2*x) |      |-\1 + \/  1 - 2*cos (2*x) / |
- I*log||---------- + --------------------------|| - I*log||---------- - --------------------------|| - I*log||- ---------- + --------------------------|| - I*log||- ---------- - --------------------------|| + arg|------------------------| + arg|------------------------| + arg|-------------------------| + arg|----------------------------|
       \|2*cos(2*x)           2*cos(2*x)        |/        \|2*cos(2*x)           2*cos(2*x)        |/        \|  2*cos(2*x)           2*cos(2*x)        |/        \|  2*cos(2*x)           2*cos(2*x)        |/      \        cos(2*x)        /      \        cos(2*x)        /      \         cos(2*x)        /      \          cos(2*x)          /

$$- i \log{\left(\left|{- \frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} - \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} - i \log{\left(\left|{- \frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} + \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} - i \log{\left(\left|{\frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} - \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} - i \log{\left(\left|{\frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} + \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{1 - \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{\cos{\left(2 x \right)}} \right)} + \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} - 1}{\cos{\left(2 x \right)}} \right)} + \arg{\left(- \frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)} + \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)}$$

producto

/       /|                      _________________|\      /       _________________\\ /       /|                        _________________|\      /        _________________\\ /       /|                        _________________|\      / /       _________________\ \\ /       /|                      _________________|\      /       _________________\\
|       ||    ___        ___   /          2      ||      |      /          2      || |       ||      ___        ___   /          2      ||      |       /          2      || |       ||      ___        ___   /          2      ||      | |      /          2      | || |       ||    ___        ___   /          2      ||      |      /          2      ||
|       ||  \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |1 - \/  1 - 2*cos (2*x) || |       ||    \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |-1 + \/  1 - 2*cos (2*x) || |       ||    \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |-\1 + \/  1 - 2*cos (2*x) / || |       ||  \/ 2       \/ 2 *\/  1 - 2*cos (2*x) ||      |1 + \/  1 - 2*cos (2*x) ||
|- I*log||---------- - --------------------------|| + arg|------------------------||*|- I*log||- ---------- + --------------------------|| + arg|-------------------------||*|- I*log||- ---------- - --------------------------|| + arg|----------------------------||*|- I*log||---------- + --------------------------|| + arg|------------------------||
\       \|2*cos(2*x)           2*cos(2*x)        |/      \        cos(2*x)        // \       \|  2*cos(2*x)           2*cos(2*x)        |/      \         cos(2*x)        // \       \|  2*cos(2*x)           2*cos(2*x)        |/      \          cos(2*x)          // \       \|2*cos(2*x)           2*cos(2*x)        |/      \        cos(2*x)        //

$$\left(- i \log{\left(\left|{- \frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} + \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{1 - \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{\cos{\left(2 x \right)}} \right)}\right) \left(- i \log{\left(\left|{\frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} - \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} - 1}{\cos{\left(2 x \right)}} \right)}\right) \left(- i \log{\left(\left|{- \frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} - \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(- \frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)}\right) \left(- i \log{\left(\left|{\frac{\sqrt{2} \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{2 \cos{\left(2 x \right)}} + \frac{\sqrt{2}}{2 \cos{\left(2 x \right)}}}\right| \right)} + \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)}\right)$$

/     /       _________________\     /              /|       _________________|\\\ /     /       _________________\     /              /|        _________________|\\\ /     /        _________________\     /              /|        _________________|\\\ /     / /       _________________\ \     /              /|       _________________|\\\
|     |      /          2      |     |              ||      /          2      |||| |     |      /          2      |     |              ||       /          2      |||| |     |       /          2      |     |              ||       /          2      |||| |     | |      /          2      | |     |              ||      /          2      ||||
|     |1 + \/  1 - 2*cos (2*x) |     |  log(2)      ||1 + \/  1 - 2*cos (2*x) |||| |     |1 - \/  1 - 2*cos (2*x) |     |  log(2)      ||-1 + \/  1 - 2*cos (2*x) |||| |     |-1 + \/  1 - 2*cos (2*x) |     |  log(2)      ||-1 + \/  1 - 2*cos (2*x) |||| |     |-\1 + \/  1 - 2*cos (2*x) / |     |  log(2)      ||1 + \/  1 - 2*cos (2*x) ||||
|- arg|------------------------| + I*|- ------ + log||------------------------||||*|- arg|------------------------| + I*|- ------ + log||-------------------------||||*|- arg|-------------------------| + I*|- ------ + log||-------------------------||||*|- arg|----------------------------| + I*|- ------ + log||------------------------||||
\     \        cos(2*x)        /     \    2         \|        cos(2*x)        |/// \     \        cos(2*x)        /     \    2         \|         cos(2*x)        |/// \     \         cos(2*x)        /     \    2         \|         cos(2*x)        |/// \     \          cos(2*x)          /     \    2         \|        cos(2*x)        |///

$$\left(i \left(\log{\left(\left|{\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} - 1}{\cos{\left(2 x \right)}}}\right| \right)} - \frac{\log{\left(2 \right)}}{2}\right) - \arg{\left(\frac{1 - \sqrt{1 - 2 \cos^{2}{\left(2 x \right)}}}{\cos{\left(2 x \right)}} \right)}\right) \left(i \left(\log{\left(\left|{\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} - 1}{\cos{\left(2 x \right)}}}\right| \right)} - \frac{\log{\left(2 \right)}}{2}\right) - \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} - 1}{\cos{\left(2 x \right)}} \right)}\right) \left(i \left(\log{\left(\left|{\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}}}\right| \right)} - \frac{\log{\left(2 \right)}}{2}\right) - \arg{\left(- \frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)}\right) \left(i \left(\log{\left(\left|{\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}}}\right| \right)} - \frac{\log{\left(2 \right)}}{2}\right) - \arg{\left(\frac{\sqrt{1 - 2 \cos^{2}{\left(2 x \right)}} + 1}{\cos{\left(2 x \right)}} \right)}\right)$$

(-arg((1 + sqrt(1 - 2*cos(2*x)^2))/cos(2*x)) + i*(-log(2)/2 + log(Abs((1 + sqrt(1 - 2*cos(2*x)^2))/cos(2*x)))))*(-arg((1 - sqrt(1 - 2*cos(2*x)^2))/cos(2*x)) + i*(-log(2)/2 + log(Abs((-1 + sqrt(1 - 2*cos(2*x)^2))/cos(2*x)))))*(-arg((-1 + sqrt(1 - 2*cos(2*x)^2))/cos(2*x)) + i*(-log(2)/2 + log(Abs((-1 + sqrt(1 - 2*cos(2*x)^2))/cos(2*x)))))*(-arg(-(1 + sqrt(1 - 2*cos(2*x)^2))/cos(2*x)) + i*(-log(2)/2 + log(Abs((1 + sqrt(1 - 2*cos(2*x)^2))/cos(2*x)))))

Otras calculadoras

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Expresiones idénticas

Expresiones con funciones

sec^2y/cos^2(2x)=2 la ecuación

Solución numérica:

Solución