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Popular Trigonometria >

10sin(x)cos^2(x)+3cos(x)=0

Solução

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

Solução

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - \frac{0.64350 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.64350 \dots}{2} + πn

Graus

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 18.43494 \dots^{\circ} + 18 0^{\circ} n, x = 108.43494 \dots^{\circ} + 18 0^{\circ} n

Passos da solução

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

Fatorar

10 sin (x) cos^{2} (x) + 3 cos (x) : cos (x) (10 sin (x) cos (x) + 3)

10 sin (x) cos^{2} (x) + 3 cos (x)

Aplicar as propriedades dos expoentes:

a^{b + c} = a^{b} a^{c}

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

= 10 cos (x) cos (x) + 3 cos (x)

Fatorar o termo comum

cos (x)

= cos (x) (10 sin (x) cos (x) + 3)

cos (x) (10 sin (x) cos (x) + 3) = 0

Resolver cada parte separadamente

cos (x) = 0 or 10 sin (x) cos (x) + 3 = 0

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

cos (x) = 0

Soluções gerais para

cos (x) = 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}

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

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

10 sin (x) cos (x) + 3 = 0 : x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

10 sin (x) cos (x) + 3 = 0

Reeecreva usando identidades trigonométricas

10 sin (x) cos (x) + 3

Utilizar a identidade trigonométrica do arco duplo:

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

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

= 3 + 10 \cdot \frac{sin ( 2 x )}{2}

3 + 10 \cdot \frac{sin ( 2 x )}{2} = 0

10 \cdot \frac{sin ( 2 x )}{2} = 5 sin (2 x)

10 \cdot \frac{sin ( 2 x )}{2}

Multiplicar frações:

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

= \frac{sin ( 2 x ) \cdot 10}{2}

Dividir:

\frac{10}{2} = 5

= 5 sin (2 x)

3 + 5 sin (2 x) = 0

Mova

3

para o lado direito

3 + 5 sin (2 x) = 0

Subtrair

3

de ambos os lados

3 + 5 sin (2 x) - 3 = 0 - 3

Simplificar

5 sin (2 x) = - 3

5 sin (2 x) = - 3

Dividir ambos os lados por

5

5 sin (2 x) = - 3

Dividir ambos os lados por

5

\frac{5 sin ( 2 x )}{5} = \frac{- 3}{5}

Simplificar

sin (2 x) = - \frac{3}{5}

sin (2 x) = - \frac{3}{5}

Aplique as propriedades trigonométricas inversas

sin (2 x) = - \frac{3}{5}

Soluções gerais para

sin (2 x) = - \frac{3}{5}

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

2 x = arcsin (- \frac{3}{5}) + 2 πn, 2 x = π + arcsin (\frac{3}{5}) + 2 πn

2 x = arcsin (- \frac{3}{5}) + 2 πn, 2 x = π + arcsin (\frac{3}{5}) + 2 πn

Resolver

2 x = arcsin (- \frac{3}{5}) + 2 πn : x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

2 x = arcsin (- \frac{3}{5}) + 2 πn

Simplificar

arcsin (- \frac{3}{5}) + 2 πn : - arcsin (\frac{3}{5}) + 2 πn

arcsin (- \frac{3}{5}) + 2 πn

Utilizar a seguinte propriedade:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{5}) = - arcsin (\frac{3}{5})

= - arcsin (\frac{3}{5}) + 2 πn

2 x = - arcsin (\frac{3}{5}) + 2 πn

Dividir ambos os lados por

2

2 x = - arcsin (\frac{3}{5}) + 2 πn

Dividir ambos os lados por

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{3}{5} )}{2} + \frac{2 πn}{2}

Simplificar

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

Resolver

2 x = π + arcsin (\frac{3}{5}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

2 x = π + arcsin (\frac{3}{5}) + 2 πn

Dividir ambos os lados por

2

2 x = π + arcsin (\frac{3}{5}) + 2 πn

Dividir ambos os lados por

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

Combinar toda as soluções

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

Mostrar soluções na forma decimal

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - \frac{0.64350 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.64350 \dots}{2} + πn

Gráfico

Exemplos populares

(2cos(x))-1=0

(2 cos (x)) - 1 = 0

3cos(x)=sin^2(x)

3 cos (x) = sin^{2} (x)

sin(θ)=(5pi)/3

sin (θ) = \frac{5 π}{3}

10cos(θ)=8

10 cos (θ) = 8

cos(θ)=(-18)/(sqrt(20)*\sqrt{26)}

cos (θ) = \frac{- 18}{20 \cdot 26}