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cos^2(x)=1-cos^2(x),[360]

Solución

cos^{2} (x) = 1 - cos^{2} (x), [36 0^{\circ}]

Solución

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

cos^{2} (x) = 1 - cos^{2} (x), [36 0^{\circ}]

Usando el método de sustitución

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

Sea:

cos (x) = u

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

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

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

Mover

u^{2}

al lado izquierdo

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

Sumar

u^{2}

a ambos lados

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

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}, [36 0^{\circ}] :

Sin solución

cos (x) = \frac{1}{2}, [36 0^{\circ}]

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

x = arccos (\frac{1}{2}) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (\frac{1}{2}) + 36 0^{\circ} n

x = arccos (\frac{1}{2}) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (\frac{1}{2}) + 36 0^{\circ} n

Soluciones para el rango

[36 0^{\circ}]

Sin soluci \overset{o}{ˊ} n

cos (x) = - \frac{1}{2}, [36 0^{\circ}] :

Sin solución

cos (x) = - \frac{1}{2}, [36 0^{\circ}]

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

cos (x) = - a \Rightarrow x = arccos (- a) + 36 0^{\circ} n, x = - arccos (- a) + 36 0^{\circ} n

x = arccos (- \frac{1}{2}) + 36 0^{\circ} n, x = - arccos (- \frac{1}{2}) + 36 0^{\circ} n

x = arccos (- \frac{1}{2}) + 36 0^{\circ} n, x = - arccos (- \frac{1}{2}) + 36 0^{\circ} n

Soluciones para el rango

[36 0^{\circ}]

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

Sin soluci \overset{o}{ˊ} n para x \in R

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