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cos^2(θ)+0.339cos(θ)-1=0

Solución

cos^{2} (θ) + 0.339 cos (θ) - 1 = 0

θ = 0.56467 \dots + 2 πn, θ = 2 π - 0.56467 \dots + 2 πn

Grados

θ = 32.35340 \dots^{\circ} + 36 0^{\circ} n, θ = 327.64659 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

cos^{2} (θ) + 0.339 cos (θ) - 1 = 0

Usando el método de sustitución

cos^{2} (θ) + 0.339 cos (θ) - 1 = 0

Sea:

cos (θ) = u

u^{2} + 0.339 u - 1 = 0

u^{2} + 0.339 u - 1 = 0 : u = \frac{- 339 + 4114921}{2000}, u = \frac{- 339 - 4114921}{2000}

u^{2} + 0.339 u - 1 = 0

Multiplicar ambos lados por

1000

u^{2} + 0.339 u - 1 = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

3

digits to the right of the decimal point, therefore multiply by

1000

u^{2} \cdot 1000 + 0.339 u \cdot 1000 - 1 \cdot 1000 = 0 \cdot 1000

Simplificar

1000 u^{2} + 339 u - 1000 = 0

1000 u^{2} + 339 u - 1000 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

1000 u^{2} + 339 u - 1000 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 1000, b = 339, c = - 1000

u_{1, 2} = \frac{- 339 \pm 33 9 ^{2} - 4 \cdot 1000 ( - 1000 )}{2 \cdot 1000}

u_{1, 2} = \frac{- 339 \pm 33 9 ^{2} - 4 \cdot 1000 ( - 1000 )}{2 \cdot 1000}

33 9^{2} - 4 \cdot 1000 (- 1000) = 4114921

33 9^{2} - 4 \cdot 1000 (- 1000)

Aplicar la regla

- (- a) = a

= 33 9^{2} + 4 \cdot 1000 \cdot 1000

Multiplicar los numeros:

4 \cdot 1000 \cdot 1000 = 4000000

= 33 9^{2} + 4000000

33 9^{2} = 114921

= 114921 + 4000000

Sumar:

114921 + 4000000 = 4114921

= 4114921

u_{1, 2} = \frac{- 339 \pm 4114921}{2 \cdot 1000}

Separar las soluciones

u_{1} = \frac{- 339 + 4114921}{2 \cdot 1000}, u_{2} = \frac{- 339 - 4114921}{2 \cdot 1000}

u = \frac{- 339 + 4114921}{2 \cdot 1000} : \frac{- 339 + 4114921}{2000}

\frac{- 339 + 4114921}{2 \cdot 1000}

Multiplicar los numeros:

2 \cdot 1000 = 2000

= \frac{- 339 + 4114921}{2000}

u = \frac{- 339 - 4114921}{2 \cdot 1000} : \frac{- 339 - 4114921}{2000}

\frac{- 339 - 4114921}{2 \cdot 1000}

Multiplicar los numeros:

2 \cdot 1000 = 2000

= \frac{- 339 - 4114921}{2000}

Las soluciones a la ecuación de segundo grado son:

u = \frac{- 339 + 4114921}{2000}, u = \frac{- 339 - 4114921}{2000}

Sustituir en la ecuación

u = cos (θ)

cos (θ) = \frac{- 339 + 4114921}{2000}, cos (θ) = \frac{- 339 - 4114921}{2000}

cos (θ) = \frac{- 339 + 4114921}{2000}, cos (θ) = \frac{- 339 - 4114921}{2000}

cos (θ) = \frac{- 339 + 4114921}{2000} : θ = arccos (\frac{- 339 + 4114921}{2000}) + 2 πn, θ = 2 π - arccos (\frac{- 339 + 4114921}{2000}) + 2 πn

cos (θ) = \frac{- 339 + 4114921}{2000}

Aplicar propiedades trigonométricas inversas

cos (θ) = \frac{- 339 + 4114921}{2000}

Soluciones generales para

cos (θ) = \frac{- 339 + 4114921}{2000}

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

θ = arccos (\frac{- 339 + 4114921}{2000}) + 2 πn, θ = 2 π - arccos (\frac{- 339 + 4114921}{2000}) + 2 πn

θ = arccos (\frac{- 339 + 4114921}{2000}) + 2 πn, θ = 2 π - arccos (\frac{- 339 + 4114921}{2000}) + 2 πn

cos (θ) = \frac{- 339 - 4114921}{2000} :

Sin solución

cos (θ) = \frac{- 339 - 4114921}{2000}

- 1 \leq cos (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

θ = arccos (\frac{- 339 + 4114921}{2000}) + 2 πn, θ = 2 π - arccos (\frac{- 339 + 4114921}{2000}) + 2 πn

Mostrar soluciones en forma decimal

θ = 0.56467 \dots + 2 πn, θ = 2 π - 0.56467 \dots + 2 πn

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

arctan (θ) = 7 \cdot \frac{540}{15}

k sin (x) - c = 0

cos (x) = sin (x - \frac{π}{2})

5 cos (x) + 2 = 0, 0 \leq x \leq 2 π