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Popular >

solvefor θ,2sin(θ)=tan(θ)

Solución

resolver para

θ, 2 sin (θ) = tan (θ)

Solución

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

Grados

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Pasos de solución

2 sin (θ) = tan (θ)

Restar

tan (θ)

de ambos lados

2 sin (θ) - tan (θ) = 0

Expresar con seno, coseno

- tan (θ) + 2 sin (θ)

Utilizar la identidad trigonométrica básica:

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

= - \frac{sin ( θ )}{cos ( θ )} + 2 sin (θ)

Simplificar

- \frac{sin ( θ )}{cos ( θ )} + 2 sin (θ) : \frac{- sin ( θ ) + 2 sin ( θ ) cos ( θ )}{cos ( θ )}

- \frac{sin ( θ )}{cos ( θ )} + 2 sin (θ)

Convertir a fracción:

2 sin (θ) = \frac{2 sin ( θ ) cos ( θ )}{cos ( θ )}

= - \frac{sin ( θ )}{cos ( θ )} + \frac{2 sin ( θ ) cos ( θ )}{cos ( θ )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- sin ( θ ) + 2 sin ( θ ) cos ( θ )}{cos ( θ )}

= \frac{- sin ( θ ) + 2 sin ( θ ) cos ( θ )}{cos ( θ )}

\frac{- sin ( θ ) + 2 cos ( θ ) sin ( θ )}{cos ( θ )} = 0

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

- sin (θ) + 2 cos (θ) sin (θ) = 0

Factorizar

- sin (θ) + 2 cos (θ) sin (θ) : sin (θ) (2 cos (θ) - 1)

- sin (θ) + 2 cos (θ) sin (θ)

Factorizar el termino común

sin (θ)

= sin (θ) (- 1 + 2 cos (θ))

sin (θ) (2 cos (θ) - 1) = 0

Resolver cada parte por separado

sin (θ) = 0 or 2 cos (θ) - 1 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

Soluciones generales para

sin (θ) = 0

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Resolver

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

2 cos (θ) - 1 = 0 : θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

2 cos (θ) - 1 = 0

Mover

1

al lado derecho

2 cos (θ) - 1 = 0

Sumar

1

a ambos lados

2 cos (θ) - 1 + 1 = 0 + 1

Simplificar

2 cos (θ) = 1

2 cos (θ) = 1

Dividir ambos lados entre

2

2 cos (θ) = 1

Dividir ambos lados entre

2

\frac{2 cos ( θ )}{2} = \frac{1}{2}

Simplificar

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

Soluciones generales para

cos (θ) = \frac{1}{2}

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}

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

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

Combinar toda las soluciones

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

Ejemplos populares

5cot^2(x)+8cot(x)+3=0

5 cot^{2} (x) + 8 cot (x) + 3 = 0

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

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

2tan^2(2x)-tan(2x)=3

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

sin(x)= 15/31

sin (x) = \frac{15}{31}

4cot(θ)-1=0

4 cot (θ) - 1 = 0