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Popular Trigonometría >

4cos^2(x)+sqrt(3)=2(sqrt(3)+1)

Solución

4 cos^{2} (x) + 3 = 2 (3 + 1)

Solución

x = 0.26179 \dots + 2 πn, x = 2 π - 0.26179 \dots + 2 πn, x = 2.87979 \dots + 2 πn, x = - 2.87979 \dots + 2 πn

Grados

x = 1 5^{\circ} + 36 0^{\circ} n, x = 34 5^{\circ} + 36 0^{\circ} n, x = 16 5^{\circ} + 36 0^{\circ} n, x = - 16 5^{\circ} + 36 0^{\circ} n

Pasos de solución

4 cos^{2} (x) + 3 = 2 (3 + 1)

Usando el método de sustitución

4 cos^{2} (x) + 3 = 2 (3 + 1)

Sea:

cos (x) = u

4 u^{2} + 3 = 2 (3 + 1)

4 u^{2} + 3 = 2 (3 + 1) : u = \frac{3 + 2}{2}, u = - \frac{3 + 2}{2}

4 u^{2} + 3 = 2 (3 + 1)

Desplace

3

a la derecha

4 u^{2} + 3 = 2 (3 + 1)

Restar

3

de ambos lados

4 u^{2} + 3 - 3 = 2 (3 + 1) - 3

Simplificar

4 u^{2} = 2 (3 + 1) - 3

4 u^{2} = 2 (3 + 1) - 3

Dividir ambos lados entre

4

4 u^{2} = 2 (3 + 1) - 3

Dividir ambos lados entre

4

\frac{4 u ^{2}}{4} = \frac{2 ( 3 + 1 )}{4} - \frac{3}{4}

Simplificar

\frac{4 u ^{2}}{4} = \frac{2 ( 3 + 1 )}{4} - \frac{3}{4}

Simplificar

\frac{4 u ^{2}}{4} : u^{2}

\frac{4 u ^{2}}{4}

Dividir:

\frac{4}{4} = 1

= u^{2}

Simplificar

\frac{2 ( 3 + 1 )}{4} - \frac{3}{4} : \frac{3 + 2}{4}

\frac{2 ( 3 + 1 )}{4} - \frac{3}{4}

Aplicar la regla

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 ( 1 + 3 ) - 3}{4}

Expandir

2 (3 + 1) - 3 : 3 + 2

2 (3 + 1) - 3

Expandir

2 (3 + 1) : 23 + 2

2 (3 + 1)

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = 2, b = 3, c = 1

= 23 + 2 \cdot 1

Multiplicar los numeros:

2 \cdot 1 = 2

= 23 + 2

= 23 + 2 - 3

Sumar elementos similares:

23 - 3 = 3

= 3 + 2

= \frac{3 + 2}{4}

u^{2} = \frac{3 + 2}{4}

u^{2} = \frac{3 + 2}{4}

u^{2} = \frac{3 + 2}{4}

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = \frac{3 + 2}{4}, u = - \frac{3 + 2}{4}

\frac{3 + 2}{4} = \frac{3 + 2}{2}

\frac{3 + 2}{4}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{3 + 2}{4}

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

= \frac{3 + 2}{2}

- \frac{3 + 2}{4} = - \frac{3 + 2}{2}

- \frac{3 + 2}{4}

Simplificar

\frac{3 + 2}{4} : \frac{3 + 2}{2}

\frac{3 + 2}{4}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{3 + 2}{4}

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

= \frac{3 + 2}{2}

= - \frac{2 + 3}{2}

= - \frac{3 + 2}{2}

u = \frac{3 + 2}{2}, u = - \frac{3 + 2}{2}

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{3 + 2}{2}, cos (x) = - \frac{3 + 2}{2}

cos (x) = \frac{3 + 2}{2}, cos (x) = - \frac{3 + 2}{2}

cos (x) = \frac{3 + 2}{2} : x = arccos (\frac{3 + 2}{2}) + 2 πn, x = 2 π - arccos (\frac{3 + 2}{2}) + 2 πn

cos (x) = \frac{3 + 2}{2}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{3 + 2}{2}

Soluciones generales para

cos (x) = \frac{3 + 2}{2}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{3 + 2}{2}) + 2 πn, x = 2 π - arccos (\frac{3 + 2}{2}) + 2 πn

x = arccos (\frac{3 + 2}{2}) + 2 πn, x = 2 π - arccos (\frac{3 + 2}{2}) + 2 πn

cos (x) = - \frac{3 + 2}{2} : x = arccos (- \frac{3 + 2}{2}) + 2 πn, x = - arccos (- \frac{3 + 2}{2}) + 2 πn

cos (x) = - \frac{3 + 2}{2}

Aplicar propiedades trigonométricas inversas

cos (x) = - \frac{3 + 2}{2}

Soluciones generales para

cos (x) = - \frac{3 + 2}{2}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{3 + 2}{2}) + 2 πn, x = - arccos (- \frac{3 + 2}{2}) + 2 πn

x = arccos (- \frac{3 + 2}{2}) + 2 πn, x = - arccos (- \frac{3 + 2}{2}) + 2 πn

Combinar toda las soluciones

x = arccos (\frac{3 + 2}{2}) + 2 πn, x = 2 π - arccos (\frac{3 + 2}{2}) + 2 πn, x = arccos (- \frac{3 + 2}{2}) + 2 πn, x = - arccos (- \frac{3 + 2}{2}) + 2 πn

Mostrar soluciones en forma decimal

x = 0.26179 \dots + 2 πn, x = 2 π - 0.26179 \dots + 2 πn, x = 2.87979 \dots + 2 πn, x = - 2.87979 \dots + 2 πn

Gráfica

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