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8tan(θ/2)+8cos(θ)tan(θ/2)=1

Solución

8 tan (\frac{θ}{2}) + 8 cos (θ) tan (\frac{θ}{2}) = 1

θ = 0.12532 \dots + 2 πn, θ = π - 0.12532 \dots + 2 πn

Pasos de solución

8 tan (\frac{θ}{2}) + 8 cos (θ) tan (\frac{θ}{2}) = 1

Restar

1

de ambos lados

8 tan (\frac{θ}{2}) + 8 cos (θ) tan (\frac{θ}{2}) - 1 = 0

Sea:

u = \frac{θ}{2}

8 tan (u) + 8 cos (2 u) tan (u) - 1 = 0

Re-escribir usando identidades trigonométricas

- 1 + 16 \cdot \frac{sin ( 2 u )}{2} = 0

16 \cdot \frac{sin ( 2 u )}{2} = 8 sin (2 u)

- 1 + 8 sin (2 u) = 0

Mover

1

al lado derecho

8 sin (2 u) = 1

Dividir ambos lados entre

8

sin (2 u) = \frac{1}{8}

Aplicar propiedades trigonométricas inversas

2 u = arcsin (\frac{1}{8}) + 2 πn, 2 u = π - arcsin (\frac{1}{8}) + 2 πn

Resolver

2 u = arcsin (\frac{1}{8}) + 2 πn : u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

Resolver

2 u = π - arcsin (\frac{1}{8}) + 2 πn : u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn, u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

Sustituir en la ecuación

u = \frac{θ}{2}

\frac{θ}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn : θ = arcsin (\frac{1}{8}) + 2 πn

\frac{θ}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn : θ = π - arcsin (\frac{1}{8}) + 2 πn

θ = arcsin (\frac{1}{8}) + 2 πn, θ = π - arcsin (\frac{1}{8}) + 2 πn

Mostrar soluciones en forma decimal

θ = 0.12532 \dots + 2 πn, θ = π - 0.12532 \dots + 2 πn

m (0) = sin (2 x) + 2

6 cos^{2} (x) + cos (x) - 1 = 0

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

6 sin^{2} (x) - 5 cos (x) - 2 = 0

sin (θ) + cos (θ) = \frac{3}{5}