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170=(160^2)/(32)sin(2θ)

Solución

170 = \frac{16 0 ^{2}}{32} sin (2 θ)

Solución

θ = \frac{0.21413 \dots}{2} + πn, θ = \frac{π}{2} - \frac{0.21413 \dots}{2} + πn

+1

Grados

θ = 6.13444 \dots^{\circ} + 18 0^{\circ} n, θ = 83.86555 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

170 = \frac{16 0 ^{2}}{32} sin (2 θ)

Intercambiar lados

\frac{16 0 ^{2}}{32} sin (2 θ) = 170

Simplificar

\frac{16 0 ^{2}}{32} : 800

\frac{16 0 ^{2}}{32}

Factorizar

16 0^{2} : 2^{10} \cdot 5^{2}

Factorizar

160 = 2^{5} \cdot 5

= (2^{5} \cdot 5)^{2}

Aplicar las leyes de los exponentes:

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

= 5^{2} (2^{5})^{2}

Simplificar

(2^{5})^{2} : 2^{10}

(2^{5})^{2}

Aplicar las leyes de los exponentes:

(a^{b})^{c} = a^{b c}

= 2^{5 \cdot 2}

Multiplicar los numeros:

5 \cdot 2 = 10

= 2^{10}

= 2^{10} \cdot 5^{2}

Factorizar

32 : 2^{5}

Factorizar

32 = 2^{5}

= \frac{2 ^{10} \cdot 5 ^{2}}{2 ^{5}}

Cancelar

\frac{2 ^{10} \cdot 5 ^{2}}{2 ^{5}} : 2^{5} \cdot 5^{2}

\frac{2 ^{10} \cdot 5 ^{2}}{2 ^{5}}

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{10}}{2 ^{5}} = 2^{10 - 5}

= 5^{2} \cdot 2^{10 - 5}

Restar:

10 - 5 = 5

= 2^{5} \cdot 5^{2}

= 2^{5} \cdot 5^{2}

2^{5} = 32

= 5^{2} \cdot 32

5^{2} = 25

= 32 \cdot 25

Multiplicar los numeros:

32 \cdot 25 = 800

= 800

800 sin (2 θ) = 170

Dividir ambos lados entre

800

800 sin (2 θ) = 170

Dividir ambos lados entre

800

\frac{800 sin ( 2 θ )}{800} = \frac{170}{800}

Simplificar

sin (2 θ) = \frac{17}{80}

sin (2 θ) = \frac{17}{80}

Aplicar propiedades trigonométricas inversas

sin (2 θ) = \frac{17}{80}

Soluciones generales para

sin (2 θ) = \frac{17}{80}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 θ = arcsin (\frac{17}{80}) + 2 πn, 2 θ = π - arcsin (\frac{17}{80}) + 2 πn

2 θ = arcsin (\frac{17}{80}) + 2 πn, 2 θ = π - arcsin (\frac{17}{80}) + 2 πn

Resolver

2 θ = arcsin (\frac{17}{80}) + 2 πn : θ = \frac{arcsin ( \frac{17}{80} )}{2} + πn

2 θ = arcsin (\frac{17}{80}) + 2 πn

Dividir ambos lados entre

2

2 θ = arcsin (\frac{17}{80}) + 2 πn

Dividir ambos lados entre

2

\frac{2 θ}{2} = \frac{arcsin ( \frac{17}{80} )}{2} + \frac{2 πn}{2}

Simplificar

θ = \frac{arcsin ( \frac{17}{80} )}{2} + πn

θ = \frac{arcsin ( \frac{17}{80} )}{2} + πn

Resolver

2 θ = π - arcsin (\frac{17}{80}) + 2 πn : θ = \frac{π}{2} - \frac{arcsin ( \frac{17}{80} )}{2} + πn

2 θ = π - arcsin (\frac{17}{80}) + 2 πn

Dividir ambos lados entre

2

2 θ = π - arcsin (\frac{17}{80}) + 2 πn

Dividir ambos lados entre

2

\frac{2 θ}{2} = \frac{π}{2} - \frac{arcsin ( \frac{17}{80} )}{2} + \frac{2 πn}{2}

Simplificar

θ = \frac{π}{2} - \frac{arcsin ( \frac{17}{80} )}{2} + πn

θ = \frac{π}{2} - \frac{arcsin ( \frac{17}{80} )}{2} + πn

θ = \frac{arcsin ( \frac{17}{80} )}{2} + πn, θ = \frac{π}{2} - \frac{arcsin ( \frac{17}{80} )}{2} + πn

Mostrar soluciones en forma decimal

θ = \frac{0.21413 \dots}{2} + πn, θ = \frac{π}{2} - \frac{0.21413 \dots}{2} + πn

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