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3(cos^2(a))/(sin^2(a))=(cot(a))^2

Solución

3 \frac{cos ^{2} ( a )}{sin ^{2} ( a )} = (cot (a))^{2}

Solución

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Grados

a = 9 0^{\circ} + 36 0^{\circ} n, a = 27 0^{\circ} + 36 0^{\circ} n

Pasos de solución

3 \cdot \frac{cos ^{2} ( a )}{sin ^{2} ( a )} = (cot (a))^{2}

Restar

(cot (a))^{2}

de ambos lados

\frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - cot^{2} (a) = 0

Simplificar

\frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - cot^{2} (a) : \frac{3 cos ^{2} ( a ) - cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

\frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - cot^{2} (a)

Convertir a fracción:

cot^{2} (a) = \frac{cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

= \frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - \frac{cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{3 cos ^{2} ( a ) - cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

\frac{3 cos ^{2} ( a ) - cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 cos^{2} (a) - cot^{2} (a) sin^{2} (a) = 0

Factorizar

3 cos^{2} (a) - cot^{2} (a) sin^{2} (a) : (3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a))

3 cos^{2} (a) - cot^{2} (a) sin^{2} (a)

Reescribir

cot^{2} (a) sin^{2} (a)

como

(cot (a) sin (a))^{2}

cot^{2} (a) sin^{2} (a)

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

cot^{2} (a) sin^{2} (a) = (cot (a) sin (a))^{2}

= (cot (a) sin (a))^{2}

= 3 cos^{2} (a) - (cot (a) sin (a))^{2}

Reescribir

3 cos^{2} (a) - (cot (a) sin (a))^{2}

como

(3 cos (a))^{2} - (cot (a) sin (a))^{2}

3 cos^{2} (a) - (cot (a) sin (a))^{2}

Aplicar las leyes de los exponentes:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} cos^{2} (a) - (cot (a) sin (a))^{2}

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

(3)^{2} cos^{2} (a) = (3 cos (a))^{2}

= (3 cos (a))^{2} - (cot (a) sin (a))^{2}

= (3 cos (a))^{2} - (cot (a) sin (a))^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

(3 cos (a))^{2} - (cot (a) sin (a))^{2} = (3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a))

= (3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a))

(3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a)) = 0

Resolver cada parte por separado

3 cos (a) + cot (a) sin (a) = 0 or 3 cos (a) - cot (a) sin (a) = 0

3 cos (a) + cot (a) sin (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

3 cos (a) + cot (a) sin (a) = 0

Re-escribir usando identidades trigonométricas

cos (a) 3 + cot (a) sin (a)

Utilizar la identidad trigonométrica básica:

cot (x) = \frac{cos ( x )}{sin ( x )}

= cos (a) 3 + \frac{cos ( a )}{sin ( a )} sin (a)

\frac{cos ( a )}{sin ( a )} sin (a) = cos (a)

\frac{cos ( a )}{sin ( a )} sin (a)

Multiplicar fracciones:

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

= \frac{cos ( a ) sin ( a )}{sin ( a )}

Eliminar los terminos comunes:

sin (a)

= cos (a)

= 3 cos (a) + cos (a)

cos (a) + cos (a) 3 = 0

Usando el método de sustitución

cos (a) + cos (a) 3 = 0

Sea:

cos (a) = u

u + u 3 = 0

u + u 3 = 0 : u = 0

u + u 3 = 0

Factorizar

u + u 3 : (1 + 3) u

u + u 3

Factorizar el termino común

u

= u (1 + 3)

(1 + 3) u = 0

Dividir ambos lados entre

1 + 3

(1 + 3) u = 0

Dividir ambos lados entre

1 + 3

\frac{( 1 + 3 ) u}{1 + 3} = \frac{0}{1 + 3}

Simplificar

u = 0

u = 0

Sustituir en la ecuación

u = cos (a)

cos (a) = 0

cos (a) = 0

cos (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

cos (a) = 0

Soluciones generales para

cos (a) = 0

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Combinar toda las soluciones

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

3 cos (a) - cot (a) sin (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

3 cos (a) - cot (a) sin (a) = 0

Re-escribir usando identidades trigonométricas

cos (a) 3 - cot (a) sin (a)

Utilizar la identidad trigonométrica básica:

cot (x) = \frac{cos ( x )}{sin ( x )}

= cos (a) 3 - \frac{cos ( a )}{sin ( a )} sin (a)

\frac{cos ( a )}{sin ( a )} sin (a) = cos (a)

\frac{cos ( a )}{sin ( a )} sin (a)

Multiplicar fracciones:

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

= \frac{cos ( a ) sin ( a )}{sin ( a )}

Eliminar los terminos comunes:

sin (a)

= cos (a)

= 3 cos (a) - cos (a)

- cos (a) + cos (a) 3 = 0

Usando el método de sustitución

- cos (a) + cos (a) 3 = 0

Sea:

cos (a) = u

- u + u 3 = 0

- u + u 3 = 0 : u = 0

- u + u 3 = 0

Factorizar

- u + u 3 : (- 1 + 3) u

- u + u 3

Factorizar el termino común

u

= u (- 1 + 3)

(- 1 + 3) u = 0

Dividir ambos lados entre

- 1 + 3

(- 1 + 3) u = 0

Dividir ambos lados entre

- 1 + 3

\frac{( - 1 + 3 ) u}{- 1 + 3} = \frac{0}{- 1 + 3}

Simplificar

u = 0

u = 0

Sustituir en la ecuación

u = cos (a)

cos (a) = 0

cos (a) = 0

cos (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

cos (a) = 0

Soluciones generales para

cos (a) = 0

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Combinar toda las soluciones

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Combinar toda las soluciones

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

Ejemplos populares

cos(x)=(-2)/(sqrt(2))

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

tan(x)=-0.22

tan (x) = - 0.22

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θ, 1 + cos (θ) = 3 cos (θ)

3sin(x)-4cos(x)=0

3 sin (x) - 4 cos (x) = 0

sin(θ)= 1/4 ,0<θ<90

sin (θ) = \frac{1}{4}, 0^{\circ} < θ < 9 0^{\circ}