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6sec(2x)+3tan(2x)-9=0

Solución

6 sec (2 x) + 3 tan (2 x) - 9 = 0

x = \frac{0.56432 \dots}{2} + πn, x = - \frac{1.20782 \dots}{2} + πn

Pasos de solución

6 sec (2 x) + 3 tan (2 x) - 9 = 0

Expresar con seno, coseno

6 \cdot \frac{1}{cos ( 2 x )} + 3 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 9 = 0

Simplificar

6 \cdot \frac{1}{cos ( 2 x )} + 3 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 9 : \frac{6 + 3 sin ( 2 x ) - 9 cos ( 2 x )}{cos ( 2 x )}

\frac{6 + 3 sin ( 2 x ) - 9 cos ( 2 x )}{cos ( 2 x )} = 0

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

6 + 3 sin (2 x) - 9 cos (2 x) = 0

Sumar

9 cos (2 x)

a ambos lados

6 + 3 sin (2 x) = 9 cos (2 x)

Elevar al cuadrado ambos lados

(6 + 3 sin (2 x))^{2} = (9 cos (2 x))^{2}

Restar

(9 cos (2 x))^{2}

de ambos lados

(6 + 3 sin (2 x))^{2} - 81 cos^{2} (2 x) = 0

Re-escribir usando identidades trigonométricas

- 45 + 36 sin (2 x) + 90 sin^{2} (2 x) = 0

Usando el método de sustitución

sin (2 x) = \frac{- 2 + 3 6}{10}, sin (2 x) = - \frac{2 + 3 6}{10}

sin (2 x) = \frac{- 2 + 3 6}{10} : x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

sin (2 x) = - \frac{2 + 3 6}{10} : x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Combinar toda las soluciones

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Verificar las soluciones sustituyendo en la ecuación original

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Mostrar soluciones en forma decimal

x = \frac{0.56432 \dots}{2} + πn, x = - \frac{1.20782 \dots}{2} + πn

3 cos^{2} (x) + 3 = 4, 0 \leq x < 2 π

tan (2 x) = 3

3 = tan (x)

sin (x) tan (x) = 3 sin (x)

12 tan (θ) + 5 = 5 tan (θ) + 5