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Popular Trigonometria >

0=1.9cos(0.1t)-1.25sin(0.2t)

Solução

0 = 1.9 cos (0.1 t) - 1.25 sin (0.2 t)

Solução

t = 5 π + 20 πn, t = 15 π + 20 πn, t = \frac{0.86331 \dots + 2 πn}{0.1}, t = \frac{π - 0.86331 \dots + 2 πn}{0.1}

Graus

t = 90 0^{\circ} + 360 0^{\circ} n, t = 270 0^{\circ} + 360 0^{\circ} n, t = 494.64197 \dots^{\circ} + 360 0^{\circ} n, t = 1305.35802 \dots^{\circ} + 360 0^{\circ} n

Passos da solução

0 = 1.9 cos (0.1 t) - 1.25 sin (0.2 t)

Trocar lados

1.9 cos (0.1 t) - 1.25 sin (0.2 t) = 0

Sea:

u = 0.1 t

1.9 cos (u) - 1.25 sin (2 u) = 0

Reeecreva usando identidades trigonométricas

- 1.25 sin (2 u) + 1.9 cos (u)

Utilizar a identidade trigonométrica do arco duplo:

sin (2 x) = 2 sin (x) cos (x)

= - 1.25 \cdot 2 sin (u) cos (u) + 1.9 cos (u)

Simplificar

= - 2.5 sin (u) cos (u) + 1.9 cos (u)

1.9 cos (u) - 2.5 cos (u) sin (u) = 0

Fatorar

1.9 cos (u) - 2.5 cos (u) sin (u) : - cos (u) (2.5 sin (u) - 1.9)

1.9 cos (u) - 2.5 cos (u) sin (u)

Fatorar o termo comum

- cos (u)

= - cos (u) (- 1.9 + 2.5 sin (u))

- cos (u) (2.5 sin (u) - 1.9) = 0

Resolver cada parte separadamente

cos (u) = 0 or 2.5 sin (u) - 1.9 = 0

cos (u) = 0 : u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = 0

Soluções gerais para

cos (u) = 0

cos (x)

tabela de periodicidade com ciclo de

2 πn

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}

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

2.5 sin (u) - 1.9 = 0 : u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

2.5 sin (u) - 1.9 = 0

Multiplicar ambos os lados por

10

2.5 sin (u) - 1.9 = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

2.5 sin (u) \cdot 10 - 1.9 \cdot 10 = 0 \cdot 10

Simplificar

25 sin (u) - 19 = 0

25 sin (u) - 19 = 0

Mova

19

para o lado direito

25 sin (u) - 19 = 0

Adicionar

19

a ambos os lados

25 sin (u) - 19 + 19 = 0 + 19

Simplificar

25 sin (u) = 19

25 sin (u) = 19

Dividir ambos os lados por

25

25 sin (u) = 19

Dividir ambos os lados por

25

\frac{25 sin ( u )}{25} = \frac{19}{25}

Simplificar

sin (u) = \frac{19}{25}

sin (u) = \frac{19}{25}

Aplique as propriedades trigonométricas inversas

sin (u) = \frac{19}{25}

Soluções gerais para

sin (u) = \frac{19}{25}

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

u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

Combinar toda as soluções

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn, u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

Substituir na equação

u = 0.1 t

0.1 t = \frac{π}{2} + 2 πn : t = 5 π + 20 πn

0.1 t = \frac{π}{2} + 2 πn

Multiplicar ambos os lados por

10

0.1 t = \frac{π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.1 t \cdot 10 = \frac{π}{2} \cdot 10 + 2 πn \cdot 10

Simplificar

t = 5 π + 20 πn

t = 5 π + 20 πn

0.1 t = \frac{3 π}{2} + 2 πn : t = 15 π + 20 πn

0.1 t = \frac{3 π}{2} + 2 πn

Multiplicar ambos os lados por

10

0.1 t = \frac{3 π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.1 t \cdot 10 = \frac{3 π}{2} \cdot 10 + 2 πn \cdot 10

Simplificar

t = 15 π + 20 πn

t = 15 π + 20 πn

0.1 t = arcsin (\frac{19}{25}) + 2 πn : t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

0.1 t = arcsin (\frac{19}{25}) + 2 πn

Dividir ambos os lados por

0.1

0.1 t = arcsin (\frac{19}{25}) + 2 πn

Dividir ambos os lados por

0.1

\frac{0.1 t}{0.1} = \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Simplificar

\frac{0.1 t}{0.1} = \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Simplificar

\frac{0.1 t}{0.1} : t

\frac{0.1 t}{0.1}

Eliminar o fator comum:

0.1

= t

Simplificar

\frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1} : \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

\frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Aplicar a regra

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

= \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

0.1 t = π - arcsin (\frac{19}{25}) + 2 πn : t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

0.1 t = π - arcsin (\frac{19}{25}) + 2 πn

Dividir ambos os lados por

0.1

0.1 t = π - arcsin (\frac{19}{25}) + 2 πn

Dividir ambos os lados por

0.1

\frac{0.1 t}{0.1} = \frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Simplificar

\frac{0.1 t}{0.1} = \frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Simplificar

\frac{0.1 t}{0.1} : t

\frac{0.1 t}{0.1}

Eliminar o fator comum:

0.1

= t

Simplificar

\frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1} : \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

\frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Aplicar a regra

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

= \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = 5 π + 20 πn, t = 15 π + 20 πn, t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}, t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

Mostrar soluções na forma decimal

t = 5 π + 20 πn, t = 15 π + 20 πn, t = \frac{0.86331 \dots + 2 πn}{0.1}, t = \frac{π - 0.86331 \dots + 2 πn}{0.1}

Gráfico

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