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1-2cos^2(x)=0,0<= x<= 2pi

Solución

1 - 2 cos^{2} (x) = 0, 0 \leq x \leq 2 π

Solución

x = \frac{π}{4}, x = \frac{7 π}{4}, x = \frac{3 π}{4}, x = \frac{5 π}{4}

+1

Grados

x = 4 5^{\circ}, x = 31 5^{\circ}, x = 13 5^{\circ}, x = 22 5^{\circ}

Pasos de solución

1 - 2 cos^{2} (x) = 0, 0 \leq x \leq 2 π

Usando el método de sustitución

1 - 2 cos^{2} (x) = 0

Sea:

cos (x) = u

1 - 2 u^{2} = 0

1 - 2 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

1 - 2 u^{2} = 0

Mover

1

al lado derecho

1 - 2 u^{2} = 0

Restar

1

de ambos lados

1 - 2 u^{2} - 1 = 0 - 1

Simplificar

- 2 u^{2} = - 1

- 2 u^{2} = - 1

Dividir ambos lados entre

- 2

- 2 u^{2} = - 1

Dividir ambos lados entre

- 2

\frac{- 2 u ^{2}}{- 2} = \frac{- 1}{- 2}

Simplificar

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

Para

x^{2} = f (a)

las soluciones son

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

u = \frac{1}{2}, u = - \frac{1}{2}

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{1}{2}, cos (x) = - \frac{1}{2}

cos (x) = \frac{1}{2}, cos (x) = - \frac{1}{2}

cos (x) = \frac{1}{2}, 0 \leq x \leq 2 π : x = \frac{π}{4}, x = \frac{7 π}{4}

cos (x) = \frac{1}{2}, 0 \leq x \leq 2 π

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{1}{2}

Soluciones generales para

cos (x) = \frac{1}{2}

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

x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn

x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn

Soluciones para el rango

0 \leq x \leq 2 π

x = \frac{π}{4}, x = \frac{7 π}{4}

cos (x) = - \frac{1}{2}, 0 \leq x \leq 2 π : x = \frac{3 π}{4}, x = \frac{5 π}{4}

cos (x) = - \frac{1}{2}, 0 \leq x \leq 2 π

Aplicar propiedades trigonométricas inversas

cos (x) = - \frac{1}{2}

Soluciones generales para

cos (x) = - \frac{1}{2}

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

x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

Soluciones para el rango

0 \leq x \leq 2 π

x = \frac{3 π}{4}, x = \frac{5 π}{4}

Combinar toda las soluciones

x = \frac{π}{4}, x = \frac{7 π}{4}, x = \frac{3 π}{4}, x = \frac{5 π}{4}

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