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Popular Trigonometría >

25sin(x)cos(x)-5sin(x)+20cos(x)=4

Solución

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) = 4

Solución

x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn, x = - 0.92729 \dots + 2 πn, x = π + 0.92729 \dots + 2 πn

+1

Grados

x = 78.46304 \dots^{\circ} + 36 0^{\circ} n, x = 281.53695 \dots^{\circ} + 36 0^{\circ} n, x = - 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 233.13010 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) = 4

Restar

4

de ambos lados

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) - 4 = 0

Factorizar

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) - 4 : (5 cos (x) - 1) (5 sin (x) + 4)

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) - 4

= (25 sin (x) cos (x) - 5 sin (x)) + (20 cos (x) - 4)

Factorizar

4

de

20 cos (x) - 4

:

4 (5 cos (x) - 1)

20 cos (x) - 4

Reescribir

20

como

5 \cdot 4

= 5 \cdot 4 cos (x) - 4

Factorizar el termino común

4

= 4 (5 cos (x) - 1)

Factorizar

5 sin (x)

de

25 sin (x) cos (x) - 5 sin (x)

:

5 sin (x) (5 cos (x) - 1)

25 sin (x) cos (x) - 5 sin (x)

Reescribir

25

como

5 \cdot 5

= 5 \cdot 5 sin (x) cos (x) - 5 sin (x)

Factorizar el termino común

5 sin (x)

= 5 sin (x) (5 cos (x) - 1)

= 4 (5 cos (x) - 1) + 5 sin (x) (5 cos (x) - 1)

Factorizar el termino común

5 cos (x) - 1

= (5 cos (x) - 1) (5 sin (x) + 4)

(5 cos (x) - 1) (5 sin (x) + 4) = 0

Resolver cada parte por separado

5 cos (x) - 1 = 0 or 5 sin (x) + 4 = 0

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

5 cos (x) - 1 = 0

Desplace

1

a la derecha

5 cos (x) - 1 = 0

Sumar

1

a ambos lados

5 cos (x) - 1 + 1 = 0 + 1

Simplificar

5 cos (x) = 1

5 cos (x) = 1

Dividir ambos lados entre

5

5 cos (x) = 1

Dividir ambos lados entre

5

\frac{5 cos ( x )}{5} = \frac{1}{5}

Simplificar

cos (x) = \frac{1}{5}

cos (x) = \frac{1}{5}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{1}{5}

Soluciones generales para

cos (x) = \frac{1}{5}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

5 sin (x) + 4 = 0 : x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

5 sin (x) + 4 = 0

Desplace

4

a la derecha

5 sin (x) + 4 = 0

Restar

4

de ambos lados

5 sin (x) + 4 - 4 = 0 - 4

Simplificar

5 sin (x) = - 4

5 sin (x) = - 4

Dividir ambos lados entre

5

5 sin (x) = - 4

Dividir ambos lados entre

5

\frac{5 sin ( x )}{5} = \frac{- 4}{5}

Simplificar

sin (x) = - \frac{4}{5}

sin (x) = - \frac{4}{5}

Aplicar propiedades trigonométricas inversas

sin (x) = - \frac{4}{5}

Soluciones generales para

sin (x) = - \frac{4}{5}

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

x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

Combinar toda las soluciones

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn, x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

Mostrar soluciones en forma decimal

x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn, x = - 0.92729 \dots + 2 πn, x = π + 0.92729 \dots + 2 πn

Gráfica

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