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2sin(x)+3cot(x)-3csc(x)=0

Solución

2 sin (x) + 3 cot (x) - 3 csc (x) = 0

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Pasos de solución

2 sin (x) + 3 cot (x) - 3 csc (x) = 0

Expresar con seno, coseno

2 sin (x) + 3 \cdot \frac{cos ( x )}{sin ( x )} - 3 \cdot \frac{1}{sin ( x )} = 0

Simplificar

2 sin (x) + 3 \cdot \frac{cos ( x )}{sin ( x )} - 3 \cdot \frac{1}{sin ( x )} : \frac{2 sin ^{2} ( x ) + 3 cos ( x ) - 3}{sin ( x )}

\frac{2 sin ^{2} ( x ) + 3 cos ( x ) - 3}{sin ( x )} = 0

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

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

Restar

3 cos (x)

de ambos lados

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

Elevar al cuadrado ambos lados

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

Restar

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

de ambos lados

(2 sin^{2} (x) - 3)^{2} - 9 cos^{2} (x) = 0

Factorizar

(2 sin^{2} (x) - 3)^{2} - 9 cos^{2} (x) : (2 sin^{2} (x) - 3 + 3 cos (x)) (2 sin^{2} (x) - 3 - 3 cos (x))

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

Resolver cada parte por separado

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

2 sin^{2} (x) - 3 + 3 cos (x) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 2 πn

2 sin^{2} (x) - 3 - 3 cos (x) = 0 : x = π + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combinar toda las soluciones

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 2 πn, x = π + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Verificar las soluciones sustituyendo en la ecuación original

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos^{2} (θ) - 3 cos (θ) + 2 = 0

cos (2 x - \frac{π}{6}) = - \frac{1}{2}, \frac{π}{2} \leq x < π

\frac{( sin ( x ) )}{( 4.2 )} = \frac{( sin ( 9 5 ^{\circ} ) )}{( 10.26 )}

- sin (x) = - cos (x)

t, f = sin (k t)