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2sin^2(3x)-3cos(3x)+1=0,0<= x<= 180

Solución

2 sin^{2} (3 x) - 3 cos (3 x) + 1 = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

Solución

x = \frac{0.81462 \dots}{3}, x = \frac{36 0 ^{\circ} - 0.81462 \dots}{3}, x = \frac{0.81462 \dots + 36 0 ^{\circ}}{3}

radianes

x = \frac{0.81462 \dots}{3}, x = \frac{2 π - 0.81462 \dots}{3}, x = \frac{0.81462 \dots + 2 π}{3}

Pasos de solución

2 sin^{2} (3 x) - 3 cos (3 x) + 1 = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

Re-escribir usando identidades trigonométricas

1 + 2 sin^{2} (3 x) - 3 cos (3 x)

Utilizar la identidad pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

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

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

Simplificar

1 + 2 (1 - cos^{2} (3 x)) - 3 cos (3 x) : - 2 cos^{2} (3 x) - 3 cos (3 x) + 3

1 + 2 (1 - cos^{2} (3 x)) - 3 cos (3 x)

Expandir

2 (1 - cos^{2} (3 x)) : 2 - 2 cos^{2} (3 x)

2 (1 - cos^{2} (3 x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 2, b = 1, c = cos^{2} (3 x)

= 2 \cdot 1 - 2 cos^{2} (3 x)

Multiplicar los numeros:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (3 x)

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

Sumar:

1 + 2 = 3

= - 2 cos^{2} (3 x) - 3 cos (3 x) + 3

= - 2 cos^{2} (3 x) - 3 cos (3 x) + 3

3 - 2 cos^{2} (3 x) - 3 cos (3 x) = 0

Usando el método de sustitución

3 - 2 cos^{2} (3 x) - 3 cos (3 x) = 0

Sea:

cos (3 x) = u

3 - 2 u^{2} - 3 u = 0

3 - 2 u^{2} - 3 u = 0 : u = - \frac{3 + 33}{4}, u = \frac{33 - 3}{4}

3 - 2 u^{2} - 3 u = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

- 2 u^{2} - 3 u + 3 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

- 2 u^{2} - 3 u + 3 = 0

Formula general para ecuaciones de segundo grado:

Para

a = - 2, b = - 3, c = 3

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

(- 3)^{2} - 4 (- 2) \cdot 3 = 33

(- 3)^{2} - 4 (- 2) \cdot 3

Aplicar la regla

- (- a) = a

= (- 3)^{2} + 4 \cdot 2 \cdot 3

Aplicar las leyes de los exponentes:

(- a)^{n} = a^{n},

n

es par

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 2 \cdot 3

Multiplicar los numeros:

4 \cdot 2 \cdot 3 = 24

= 3^{2} + 24

3^{2} = 9

= 9 + 24

Sumar:

9 + 24 = 33

= 33

u_{1, 2} = \frac{- ( - 3 ) \pm 33}{2 ( - 2 )}

Separar las soluciones

u_{1} = \frac{- ( - 3 ) + 33}{2 ( - 2 )}, u_{2} = \frac{- ( - 3 ) - 33}{2 ( - 2 )}

u = \frac{- ( - 3 ) + 33}{2 ( - 2 )} : - \frac{3 + 33}{4}

\frac{- ( - 3 ) + 33}{2 ( - 2 )}

Quitar los parentesis:

(- a) = - a, - (- a) = a

= \frac{3 + 33}{- 2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{3 + 33}{- 4}

Aplicar las propiedades de las fracciones:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{3 + 33}{4}

u = \frac{- ( - 3 ) - 33}{2 ( - 2 )} : \frac{33 - 3}{4}

\frac{- ( - 3 ) - 33}{2 ( - 2 )}

Quitar los parentesis:

(- a) = - a, - (- a) = a

= \frac{3 - 33}{- 2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{3 - 33}{- 4}

Aplicar las propiedades de las fracciones:

\frac{- a}{- b} = \frac{a}{b}

3 - 33 = - (33 - 3)

= \frac{33 - 3}{4}

Las soluciones a la ecuación de segundo grado son:

u = - \frac{3 + 33}{4}, u = \frac{33 - 3}{4}

Sustituir en la ecuación

u = cos (3 x)

cos (3 x) = - \frac{3 + 33}{4}, cos (3 x) = \frac{33 - 3}{4}

cos (3 x) = - \frac{3 + 33}{4}, cos (3 x) = \frac{33 - 3}{4}

cos (3 x) = - \frac{3 + 33}{4}, 0 \leq x \leq 18 0^{\circ} :

Sin solución

cos (3 x) = - \frac{3 + 33}{4}, 0 \leq x \leq 18 0^{\circ}

- 1 \leq cos (x) \leq 1

Sin soluci \overset{o}{ˊ} n

cos (3 x) = \frac{33 - 3}{4}, 0 \leq x \leq 18 0^{\circ} : x = \frac{arccos ( \frac{33 - 3}{4} )}{3}, x = \frac{36 0 ^{\circ} - arccos ( \frac{33 - 3}{4} )}{3}, x = \frac{arccos ( \frac{33 - 3}{4} ) + 36 0 ^{\circ}}{3}

cos (3 x) = \frac{33 - 3}{4}, 0 \leq x \leq 18 0^{\circ}

Aplicar propiedades trigonométricas inversas

cos (3 x) = \frac{33 - 3}{4}

Soluciones generales para

cos (3 x) = \frac{33 - 3}{4}

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

3 x = arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n, 3 x = 36 0^{\circ} - arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n

3 x = arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n, 3 x = 36 0^{\circ} - arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n

Resolver

3 x = arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n : x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}

3 x = arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n

Dividir ambos lados entre

3

3 x = arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n

Dividir ambos lados entre

3

\frac{3 x}{3} = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}

Simplificar

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}

Resolver

3 x = 36 0^{\circ} - arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n : x = 12 0^{\circ} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}

3 x = 36 0^{\circ} - arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n

Dividir ambos lados entre

3

3 x = 36 0^{\circ} - arccos (\frac{33 - 3}{4}) + 36 0^{\circ} n

Dividir ambos lados entre

3

\frac{3 x}{3} = 12 0^{\circ} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}

Simplificar

x = 12 0^{\circ} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}

x = 12 0^{\circ} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}, x = 12 0^{\circ} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{36 0 ^{\circ} n}{3}

Soluciones para el rango

0 \leq x \leq 18 0^{\circ}

x = \frac{arccos ( \frac{33 - 3}{4} )}{3}, x = \frac{36 0 ^{\circ} - arccos ( \frac{33 - 3}{4} )}{3}, x = \frac{arccos ( \frac{33 - 3}{4} ) + 36 0 ^{\circ}}{3}

Combinar toda las soluciones

x = \frac{arccos ( \frac{33 - 3}{4} )}{3}, x = \frac{36 0 ^{\circ} - arccos ( \frac{33 - 3}{4} )}{3}, x = \frac{arccos ( \frac{33 - 3}{4} ) + 36 0 ^{\circ}}{3}

Mostrar soluciones en forma decimal

x = \frac{0.81462 \dots}{3}, x = \frac{36 0 ^{\circ} - 0.81462 \dots}{3}, x = \frac{0.81462 \dots + 36 0 ^{\circ}}{3}

Ejemplos populares

2sin(x/2)=1-cos(x)

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

sin(x)=(-sin(2x))/2

sin (x) = \frac{- sin ( 2 x )}{2}

12sin(A)+4=4sin(A)+4

12 sin (A) + 4 = 4 sin (A) + 4

2sin^2(x)=3-3cos(x)

2 sin^{2} (x) = 3 - 3 cos (x)

cos(x)+sin(x)tan(x)=1

cos (x) + sin (x) tan (x) = 1