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Popular Trigonometria >

sec^{22}(x)=1-tan^2(x)

Solução

sec^{22} (x) = 1 - tan^{2} (x)

Solução

x = 2 πn, x = π + 2 πn

Graus

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Passos da solução

sec^{22} (x) = 1 - tan^{2} (x)

Subtrair

1 - tan^{2} (x)

de ambos os lados

sec^{22} (x) - 1 + tan^{2} (x) = 0

Reeecreva usando identidades trigonométricas

- 1 + sec^{22} (x) + tan^{2} (x)

Utilizar a identidade trigonométrica pitagórica:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= - 1 + sec^{22} (x) + sec^{2} (x) - 1

Simplificar

- 1 + sec^{22} (x) + sec^{2} (x) - 1 : sec^{22} (x) + sec^{2} (x) - 2

- 1 + sec^{22} (x) + sec^{2} (x) - 1

Agrupar termos semelhantes

= sec^{22} (x) + sec^{2} (x) - 1 - 1

Subtrair:

- 1 - 1 = - 2

= sec^{22} (x) + sec^{2} (x) - 2

= sec^{22} (x) + sec^{2} (x) - 2

- 2 + sec^{22} (x) + sec^{2} (x) = 0

Usando o método de substituição

- 2 + sec^{22} (x) + sec^{2} (x) = 0

Sea:

sec (x) = u

- 2 + u^{22} + u^{2} = 0

- 2 + u^{22} + u^{2} = 0 : u = 1, u = - 1

- 2 + u^{22} + u^{2} = 0

Escrever na forma padrão

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{22} + u^{2} - 2 = 0

Reescrever a equação com

v = u^{2}

v^{11} = u^{22}

v^{11} + v - 2 = 0

Resolver

v^{11} + v - 2 = 0 : v = 1

v^{11} + v - 2 = 0

Fatorar

v^{11} + v - 2 : (v - 1) (v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2)

v^{11} + v - 2

Utilizar o teorema das raízes racionais

a_{0} = 2, a_{n} = 1

Os divisores de

a_{0} : 1, 2,

Os divisores de

a_{n} : 1

Portanto, verificar os seguintes números racionais:

\pm \frac{1 , 2}{1}

\frac{1}{1}

é a raiz da expressão, portanto, fatorar

v - 1

= (v - 1) \frac{v ^{11} + v - 2}{v - 1}

\frac{v ^{11} + v - 2}{v - 1} = v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2

\frac{v ^{11} + v - 2}{v - 1}

Dividir

\frac{v ^{11} + v - 2}{v - 1} : \frac{v ^{11} + v - 2}{v - 1} = v^{10} + \frac{v ^{10} + v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{11} + v - 2

e o divisor

v - 1 : \frac{v ^{11}}{v} = v^{10}

Quociente = v^{10}

Multiplicar

v - 1

por

v^{10} : v^{11} - v^{10}

Subtrair

v^{11} - v^{10}

v^{11} + v - 2

para obter um novo resto

Resto = v^{10} + v - 2

Portanto

\frac{v ^{11} + v - 2}{v - 1} = v^{10} + \frac{v ^{10} + v - 2}{v - 1}

= v^{10} + \frac{v ^{10} + v - 2}{v - 1}

Dividir

\frac{v ^{10} + v - 2}{v - 1} : \frac{v ^{10} + v - 2}{v - 1} = v^{9} + \frac{v ^{9} + v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{10} + v - 2

e o divisor

v - 1 : \frac{v ^{10}}{v} = v^{9}

Quociente = v^{9}

Multiplicar

v - 1

por

v^{9} : v^{10} - v^{9}

Subtrair

v^{10} - v^{9}

v^{10} + v - 2

para obter um novo resto

Resto = v^{9} + v - 2

Portanto

\frac{v ^{10} + v - 2}{v - 1} = v^{9} + \frac{v ^{9} + v - 2}{v - 1}

= v^{10} + v^{9} + \frac{v ^{9} + v - 2}{v - 1}

Dividir

\frac{v ^{9} + v - 2}{v - 1} : \frac{v ^{9} + v - 2}{v - 1} = v^{8} + \frac{v ^{8} + v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{9} + v - 2

e o divisor

v - 1 : \frac{v ^{9}}{v} = v^{8}

Quociente = v^{8}

Multiplicar

v - 1

por

v^{8} : v^{9} - v^{8}

Subtrair

v^{9} - v^{8}

v^{9} + v - 2

para obter um novo resto

Resto = v^{8} + v - 2

Portanto

\frac{v ^{9} + v - 2}{v - 1} = v^{8} + \frac{v ^{8} + v - 2}{v - 1}

= v^{10} + v^{9} + v^{8} + \frac{v ^{8} + v - 2}{v - 1}

Dividir

\frac{v ^{8} + v - 2}{v - 1} : \frac{v ^{8} + v - 2}{v - 1} = v^{7} + \frac{v ^{7} + v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{8} + v - 2

e o divisor

v - 1 : \frac{v ^{8}}{v} = v^{7}

Quociente = v^{7}

Multiplicar

v - 1

por

v^{7} : v^{8} - v^{7}

Subtrair

v^{8} - v^{7}

v^{8} + v - 2

para obter um novo resto

Resto = v^{7} + v - 2

Portanto

\frac{v ^{8} + v - 2}{v - 1} = v^{7} + \frac{v ^{7} + v - 2}{v - 1}

= v^{10} + v^{9} + v^{8} + v^{7} + \frac{v ^{7} + v - 2}{v - 1}

Dividir

\frac{v ^{7} + v - 2}{v - 1} : \frac{v ^{7} + v - 2}{v - 1} = v^{6} + \frac{v ^{6} + v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{7} + v - 2

e o divisor

v - 1 : \frac{v ^{7}}{v} = v^{6}

Quociente = v^{6}

Multiplicar

v - 1

por

v^{6} : v^{7} - v^{6}

Subtrair

v^{7} - v^{6}

v^{7} + v - 2

para obter um novo resto

Resto = v^{6} + v - 2

Portanto

\frac{v ^{7} + v - 2}{v - 1} = v^{6} + \frac{v ^{6} + v - 2}{v - 1}

= v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + \frac{v ^{6} + v - 2}{v - 1}

Dividir

\frac{v ^{6} + v - 2}{v - 1} : \frac{v ^{6} + v - 2}{v - 1} = v^{5} + \frac{v ^{5} + v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{6} + v - 2

e o divisor

v - 1 : \frac{v ^{6}}{v} = v^{5}

Quociente = v^{5}

Multiplicar

v - 1

por

v^{5} : v^{6} - v^{5}

Subtrair

v^{6} - v^{5}

v^{6} + v - 2

para obter um novo resto

Resto = v^{5} + v - 2

Portanto

\frac{v ^{6} + v - 2}{v - 1} = v^{5} + \frac{v ^{5} + v - 2}{v - 1}

= v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + \frac{v ^{5} + v - 2}{v - 1}

Dividir

\frac{v ^{5} + v - 2}{v - 1} : \frac{v ^{5} + v - 2}{v - 1} = v^{4} + \frac{v ^{4} + v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{5} + v - 2

e o divisor

v - 1 : \frac{v ^{5}}{v} = v^{4}

Quociente = v^{4}

Multiplicar

v - 1

por

v^{4} : v^{5} - v^{4}

Subtrair

v^{5} - v^{4}

v^{5} + v - 2

para obter um novo resto

Resto = v^{4} + v - 2

Portanto

\frac{v ^{5} + v - 2}{v - 1} = v^{4} + \frac{v ^{4} + v - 2}{v - 1}

= v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + \frac{v ^{4} + v - 2}{v - 1}

Dividir

\frac{v ^{4} + v - 2}{v - 1} : \frac{v ^{4} + v - 2}{v - 1} = v^{3} + \frac{v ^{3} + v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{4} + v - 2

e o divisor

v - 1 : \frac{v ^{4}}{v} = v^{3}

Quociente = v^{3}

Multiplicar

v - 1

por

v^{3} : v^{4} - v^{3}

Subtrair

v^{4} - v^{3}

v^{4} + v - 2

para obter um novo resto

Resto = v^{3} + v - 2

Portanto

\frac{v ^{4} + v - 2}{v - 1} = v^{3} + \frac{v ^{3} + v - 2}{v - 1}

= v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + \frac{v ^{3} + v - 2}{v - 1}

Dividir

\frac{v ^{3} + v - 2}{v - 1} : \frac{v ^{3} + v - 2}{v - 1} = v^{2} + \frac{v ^{2} + v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{3} + v - 2

e o divisor

v - 1 : \frac{v ^{3}}{v} = v^{2}

Quociente = v^{2}

Multiplicar

v - 1

por

v^{2} : v^{3} - v^{2}

Subtrair

v^{3} - v^{2}

v^{3} + v - 2

para obter um novo resto

Resto = v^{2} + v - 2

Portanto

\frac{v ^{3} + v - 2}{v - 1} = v^{2} + \frac{v ^{2} + v - 2}{v - 1}

= v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + \frac{v ^{2} + v - 2}{v - 1}

Dividir

\frac{v ^{2} + v - 2}{v - 1} : \frac{v ^{2} + v - 2}{v - 1} = v + \frac{2 v - 2}{v - 1}

Dividir os coeficientes dos termos de maior grau do numerador

v^{2} + v - 2

e o divisor

v - 1 : \frac{v ^{2}}{v} = v

Quociente = v

Multiplicar

v - 1

por

v : v^{2} - v

Subtrair

v^{2} - v

v^{2} + v - 2

para obter um novo resto

Resto = 2 v - 2

Portanto

\frac{v ^{2} + v - 2}{v - 1} = v + \frac{2 v - 2}{v - 1}

= v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + \frac{2 v - 2}{v - 1}

Dividir

\frac{2 v - 2}{v - 1} : \frac{2 v - 2}{v - 1} = 2

Dividir os coeficientes dos termos de maior grau do numerador

2 v - 2

e o divisor

v - 1 : \frac{2 v}{v} = 2

Quociente = 2

Multiplicar

v - 1

por

2 : 2 v - 2

Subtrair

2 v - 2

2 v - 2

para obter um novo resto

Resto = 0

Portanto

\frac{2 v - 2}{v - 1} = 2

= v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2

= (v - 1) (v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2)

(v - 1) (v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2) = 0

Usando o princípio do fator zero: Se

ab = 0

então

a = 0

b = 0

v - 1 = 0 or v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2 = 0

Resolver

v - 1 = 0 : v = 1

v - 1 = 0

Mova

1

para o lado direito

v - 1 = 0

Adicionar

1

a ambos os lados

v - 1 + 1 = 0 + 1

Simplificar

v = 1

v = 1

Resolver

v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2 = 0 :

Sem solução para

v \in R

v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2 = 0

Encontrar uma solução para

v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2 = 0

utilizando o método de Newton-Raphson:

Sem solução para

v \in R

v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2 = 0

Definição de método de Newton-Raphson

f (v) = v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2

Encontrar

f^{^{'}} (v) : 10 v^{9} + 9 v^{8} + 8 v^{7} + 7 v^{6} + 6 v^{5} + 5 v^{4} + 4 v^{3} + 3 v^{2} + 2 v + 1

\frac{d}{d v} (v^{10} + v^{9} + v^{8} + v^{7} + v^{6} + v^{5} + v^{4} + v^{3} + v^{2} + v + 2)

Aplicar a regra da soma/diferença:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d v} (v^{10}) + \frac{d}{d v} (v^{9}) + \frac{d}{d v} (v^{8}) + \frac{d}{d v} (v^{7}) + \frac{d}{d v} (v^{6}) + \frac{d}{d v} (v^{5}) + \frac{d}{d v} (v^{4}) + \frac{d}{d v} (v^{3}) + \frac{d}{d v} (v^{2}) + \frac{d v}{d v} + \frac{d}{d v} (2)

\frac{d}{d v} (v^{10}) = 10 v^{9}

\frac{d}{d v} (v^{10})

Aplicar a regra da potência:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 10 v^{10 - 1}

Simplificar

= 10 v^{9}

\frac{d}{d v} (v^{9}) = 9 v^{8}

\frac{d}{d v} (v^{9})

Aplicar a regra da potência:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 9 v^{9 - 1}

Simplificar

= 9 v^{8}

\frac{d}{d v} (v^{8}) = 8 v^{7}

\frac{d}{d v} (v^{8})

Aplicar a regra da potência:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 8 v^{8 - 1}

Simplificar

= 8 v^{7}

\frac{d}{d v} (v^{7}) = 7 v^{6}

\frac{d}{d v} (v^{7})

Aplicar a regra da potência:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 7 v^{7 - 1}

Simplificar

= 7 v^{6}

\frac{d}{d v} (v^{6}) = 6 v^{5}

\frac{d}{d v} (v^{6})

Aplicar a regra da potência:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 6 v^{6 - 1}

Simplificar

= 6 v^{5}

\frac{d}{d v} (v^{5}) = 5 v^{4}

\frac{d}{d v} (v^{5})

Aplicar a regra da potência:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 5 v^{5 - 1}

Simplificar

= 5 v^{4}

\frac{d}{d v} (v^{4}) = 4 v^{3}

\frac{d}{d v} (v^{4})

Aplicar a regra da potência:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 v^{4 - 1}

Simplificar

= 4 v^{3}

\frac{d}{d v} (v^{3}) = 3 v^{2}

\frac{d}{d v} (v^{3})

Aplicar a regra da potência:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 v^{3 - 1}

Simplificar

= 3 v^{2}

\frac{d}{d v} (v^{2}) = 2 v

\frac{d}{d v} (v^{2})

Aplicar a regra da potência:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 v^{2 - 1}

Simplificar

= 2 v

\frac{d v}{d v} = 1

\frac{d v}{d v}

Aplicar a regra da derivação:

\frac{d v}{d v} = 1

= 1

\frac{d}{d v} (2) = 0

\frac{d}{d v} (2)

Derivada de uma constante:

\frac{d}{d x} (a) = 0

= 0

= 10 v^{9} + 9 v^{8} + 8 v^{7} + 7 v^{6} + 6 v^{5} + 5 v^{4} + 4 v^{3} + 3 v^{2} + 2 v + 1 + 0

Simplificar

= 10 v^{9} + 9 v^{8} + 8 v^{7} + 7 v^{6} + 6 v^{5} + 5 v^{4} + 4 v^{3} + 3 v^{2} + 2 v + 1

Seja

v_{0} = - 2

Calcular

v_{n + 1}

até que

Δ v_{n + 1} < 0.000001

v_{1} = - 1.80606 \dots : Δ v_{1} = 0.19393 \dots

f (v_{0}) = (- 2)^{10} + (- 2)^{9} + (- 2)^{8} + (- 2)^{7} + (- 2)^{6} + (- 2)^{5} + (- 2)^{4} + (- 2)^{3} + (- 2)^{2} + (- 2) + 2 = 684

f^{^{'}} (v_{0}) = 10 (- 2)^{9} + 9 (- 2)^{8} + 8 (- 2)^{7} + 7 (- 2)^{6} + 6 (- 2)^{5} + 5 (- 2)^{4} + 4 (- 2)^{3} + 3 (- 2)^{2} + 2 (- 2) + 1 = - 3527

v_{1} = - 1.80606 \dots

Δ v_{1} = ∣ - 1.80606 \dots - (- 2) ∣ = 0.19393 \dots

Δ v_{1} = 0.19393 \dots

v_{2} = - 1.63066 \dots : Δ v_{2} = 0.17540 \dots

f (v_{1}) = (- 1.80606 \dots)^{10} + (- 1.80606 \dots)^{9} + (- 1.80606 \dots)^{8} + (- 1.80606 \dots)^{7} + (- 1.80606 \dots)^{6} + (- 1.80606 \dots)^{5} + (- 1.80606 \dots)^{4} + (- 1.80606 \dots)^{3} + (- 1.80606 \dots)^{2} + (- 1.80606 \dots) + 2 = 239.02703 \dots

f^{^{'}} (v_{1}) = 10 (- 1.80606 \dots)^{9} + 9 (- 1.80606 \dots)^{8} + 8 (- 1.80606 \dots)^{7} + 7 (- 1.80606 \dots)^{6} + 6 (- 1.80606 \dots)^{5} + 5 (- 1.80606 \dots)^{4} + 4 (- 1.80606 \dots)^{3} + 3 (- 1.80606 \dots)^{2} + 2 (- 1.80606 \dots) + 1 = - 1362.72658 \dots

v_{2} = - 1.63066 \dots

Δ v_{2} = ∣ - 1.63066 \dots - (- 1.80606 \dots) ∣ = 0.17540 \dots

Δ v_{2} = 0.17540 \dots

v_{3} = - 1.47089 \dots : Δ v_{3} = 0.15976 \dots

f (v_{2}) = (- 1.63066 \dots)^{10} + (- 1.63066 \dots)^{9} + (- 1.63066 \dots)^{8} + (- 1.63066 \dots)^{7} + (- 1.63066 \dots)^{6} + (- 1.63066 \dots)^{5} + (- 1.63066 \dots)^{4} + (- 1.63066 \dots)^{3} + (- 1.63066 \dots)^{2} + (- 1.63066 \dots) + 2 = 83.78328 \dots

f^{^{'}} (v_{2}) = 10 (- 1.63066 \dots)^{9} + 9 (- 1.63066 \dots)^{8} + 8 (- 1.63066 \dots)^{7} + 7 (- 1.63066 \dots)^{6} + 6 (- 1.63066 \dots)^{5} + 5 (- 1.63066 \dots)^{4} + 4 (- 1.63066 \dots)^{3} + 3 (- 1.63066 \dots)^{2} + 2 (- 1.63066 \dots) + 1 = - 524.39987 \dots

v_{3} = - 1.47089 \dots

Δ v_{3} = ∣ - 1.47089 \dots - (- 1.63066 \dots) ∣ = 0.15976 \dots

Δ v_{3} = 0.15976 \dots

v_{4} = - 1.32236 \dots : Δ v_{4} = 0.14852 \dots

f (v_{3}) = (- 1.47089 \dots)^{10} + (- 1.47089 \dots)^{9} + (- 1.47089 \dots)^{8} + (- 1.47089 \dots)^{7} + (- 1.47089 \dots)^{6} + (- 1.47089 \dots)^{5} + (- 1.47089 \dots)^{4} + (- 1.47089 \dots)^{3} + (- 1.47089 \dots)^{2} + (- 1.47089 \dots) + 2 = 29.62369 \dots

f^{^{'}} (v_{3}) = 10 (- 1.47089 \dots)^{9} + 9 (- 1.47089 \dots)^{8} + 8 (- 1.47089 \dots)^{7} + 7 (- 1.47089 \dots)^{6} + 6 (- 1.47089 \dots)^{5} + 5 (- 1.47089 \dots)^{4} + 4 (- 1.47089 \dots)^{3} + 3 (- 1.47089 \dots)^{2} + 2 (- 1.47089 \dots) + 1 = - 199.44976 \dots

v_{4} = - 1.32236 \dots

Δ v_{4} = ∣ - 1.32236 \dots - (- 1.47089 \dots) ∣ = 0.14852 \dots

Δ v_{4} = 0.14852 \dots

v_{5} = - 1.17573 \dots : Δ v_{5} = 0.14662 \dots

f (v_{4}) = (- 1.32236 \dots)^{10} + (- 1.32236 \dots)^{9} + (- 1.32236 \dots)^{8} + (- 1.32236 \dots)^{7} + (- 1.32236 \dots)^{6} + (- 1.32236 \dots)^{5} + (- 1.32236 \dots)^{4} + (- 1.32236 \dots)^{3} + (- 1.32236 \dots)^{2} + (- 1.32236 \dots) + 2 = 10.74041 \dots

f^{^{'}} (v_{4}) = 10 (- 1.32236 \dots)^{9} + 9 (- 1.32236 \dots)^{8} + 8 (- 1.32236 \dots)^{7} + 7 (- 1.32236 \dots)^{6} + 6 (- 1.32236 \dots)^{5} + 5 (- 1.32236 \dots)^{4} + 4 (- 1.32236 \dots)^{3} + 3 (- 1.32236 \dots)^{2} + 2 (- 1.32236 \dots) + 1 = - 73.24877 \dots

v_{5} = - 1.17573 \dots

Δ v_{5} = ∣ - 1.17573 \dots - (- 1.32236 \dots) ∣ = 0.14662 \dots

Δ v_{5} = 0.14662 \dots

v_{6} = - 1.00166 \dots : Δ v_{6} = 0.17407 \dots

f (v_{5}) = (- 1.17573 \dots)^{10} + (- 1.17573 \dots)^{9} + (- 1.17573 \dots)^{8} + (- 1.17573 \dots)^{7} + (- 1.17573 \dots)^{6} + (- 1.17573 \dots)^{5} + (- 1.17573 \dots)^{4} + (- 1.17573 \dots)^{3} + (- 1.17573 \dots)^{2} + (- 1.17573 \dots) + 2 = 4.18738 \dots

f^{^{'}} (v_{5}) = 10 (- 1.17573 \dots)^{9} + 9 (- 1.17573 \dots)^{8} + 8 (- 1.17573 \dots)^{7} + 7 (- 1.17573 \dots)^{6} + 6 (- 1.17573 \dots)^{5} + 5 (- 1.17573 \dots)^{4} + 4 (- 1.17573 \dots)^{3} + 3 (- 1.17573 \dots)^{2} + 2 (- 1.17573 \dots) + 1 = - 24.05562 \dots

v_{6} = - 1.00166 \dots

Δ v_{6} = ∣ - 1.00166 \dots - (- 1.17573 \dots) ∣ = 0.17407 \dots

Δ v_{6} = 0.17407 \dots

v_{7} = - 0.60661 \dots : Δ v_{7} = 0.39505 \dots

f (v_{6}) = (- 1.00166 \dots)^{10} + (- 1.00166 \dots)^{9} + (- 1.00166 \dots)^{8} + (- 1.00166 \dots)^{7} + (- 1.00166 \dots)^{6} + (- 1.00166 \dots)^{5} + (- 1.00166 \dots)^{4} + (- 1.00166 \dots)^{3} + (- 1.00166 \dots)^{2} + (- 1.00166 \dots) + 2 = 2.00840 \dots

f^{^{'}} (v_{6}) = 10 (- 1.00166 \dots)^{9} + 9 (- 1.00166 \dots)^{8} + 8 (- 1.00166 \dots)^{7} + 7 (- 1.00166 \dots)^{6} + 6 (- 1.00166 \dots)^{5} + 5 (- 1.00166 \dots)^{4} + 4 (- 1.00166 \dots)^{3} + 3 (- 1.00166 \dots)^{2} + 2 (- 1.00166 \dots) + 1 = - 5.08391 \dots

v_{7} = - 0.60661 \dots

Δ v_{7} = ∣ - 0.60661 \dots - (- 1.00166 \dots) ∣ = 0.39505 \dots

Δ v_{7} = 0.39505 \dots

v_{8} = - 5.34688 \dots : Δ v_{8} = 4.74026 \dots

f (v_{7}) = (- 0.60661 \dots)^{10} + (- 0.60661 \dots)^{9} + (- 0.60661 \dots)^{8} + (- 0.60661 \dots)^{7} + (- 0.60661 \dots)^{6} + (- 0.60661 \dots)^{5} + (- 0.60661 \dots)^{4} + (- 0.60661 \dots)^{3} + (- 0.60661 \dots)^{2} + (- 0.60661 \dots) + 2 = 1.62497 \dots

f^{^{'}} (v_{7}) = 10 (- 0.60661 \dots)^{9} + 9 (- 0.60661 \dots)^{8} + 8 (- 0.60661 \dots)^{7} + 7 (- 0.60661 \dots)^{6} + 6 (- 0.60661 \dots)^{5} + 5 (- 0.60661 \dots)^{4} + 4 (- 0.60661 \dots)^{3} + 3 (- 0.60661 \dots)^{2} + 2 (- 0.60661 \dots) + 1 = 0.34280 \dots

v_{8} = - 5.34688 \dots

Δ v_{8} = ∣ - 5.34688 \dots - (- 0.60661 \dots) ∣ = 4.74026 \dots

Δ v_{8} = 4.74026 \dots

v_{9} = - 4.82048 \dots : Δ v_{9} = 0.52639 \dots

f (v_{8}) = (- 5.34688 \dots)^{10} + (- 5.34688 \dots)^{9} + (- 5.34688 \dots)^{8} + (- 5.34688 \dots)^{7} + (- 5.34688 \dots)^{6} + (- 5.34688 \dots)^{5} + (- 5.34688 \dots)^{4} + (- 5.34688 \dots)^{3} + (- 5.34688 \dots)^{2} + (- 5.34688 \dots) + 2 = 16089690.13941 \dots

f^{^{'}} (v_{8}) = 10 (- 5.34688 \dots)^{9} + 9 (- 5.34688 \dots)^{8} + 8 (- 5.34688 \dots)^{7} + 7 (- 5.34688 \dots)^{6} + 6 (- 5.34688 \dots)^{5} + 5 (- 5.34688 \dots)^{4} + 4 (- 5.34688 \dots)^{3} + 3 (- 5.34688 \dots)^{2} + 2 (- 5.34688 \dots) + 1 = - 30565830.39717 \dots

v_{9} = - 4.82048 \dots

Δ v_{9} = ∣ - 4.82048 \dots - (- 5.34688 \dots) ∣ = 0.52639 \dots

Δ v_{9} = 0.52639 \dots

v_{10} = - 4.34658 \dots : Δ v_{10} = 0.47390 \dots

f (v_{9}) = (- 4.82048 \dots)^{10} + (- 4.82048 \dots)^{9} + (- 4.82048 \dots)^{8} + (- 4.82048 \dots)^{7} + (- 4.82048 \dots)^{6} + (- 4.82048 \dots)^{5} + (- 4.82048 \dots)^{4} + (- 4.82048 \dots)^{3} + (- 4.82048 \dots)^{2} + (- 4.82048 \dots) + 2 = 5611032.52684 \dots

f^{^{'}} (v_{9}) = 10 (- 4.82048 \dots)^{9} + 9 (- 4.82048 \dots)^{8} + 8 (- 4.82048 \dots)^{7} + 7 (- 4.82048 \dots)^{6} + 6 (- 4.82048 \dots)^{5} + 5 (- 4.82048 \dots)^{4} + 4 (- 4.82048 \dots)^{3} + 3 (- 4.82048 \dots)^{2} + 2 (- 4.82048 \dots) + 1 = - 11839945.22082 \dots

v_{10} = - 4.34658 \dots

Δ v_{10} = ∣ - 4.34658 \dots - (- 4.82048 \dots) ∣ = 0.47390 \dots

Δ v_{10} = 0.47390 \dots

v_{11} = - 3.91990 \dots : Δ v_{11} = 0.42667 \dots

f (v_{10}) = (- 4.34658 \dots)^{10} + (- 4.34658 \dots)^{9} + (- 4.34658 \dots)^{8} + (- 4.34658 \dots)^{7} + (- 4.34658 \dots)^{6} + (- 4.34658 \dots)^{5} + (- 4.34658 \dots)^{4} + (- 4.34658 \dots)^{3} + (- 4.34658 \dots)^{2} + (- 4.34658 \dots) + 2 = 1956819.81301 \dots

f^{^{'}} (v_{10}) = 10 (- 4.34658 \dots)^{9} + 9 (- 4.34658 \dots)^{8} + 8 (- 4.34658 \dots)^{7} + 7 (- 4.34658 \dots)^{6} + 6 (- 4.34658 \dots)^{5} + 5 (- 4.34658 \dots)^{4} + 4 (- 4.34658 \dots)^{3} + 3 (- 4.34658 \dots)^{2} + 2 (- 4.34658 \dots) + 1 = - 4586173.62069 \dots

v_{11} = - 3.91990 \dots

Δ v_{11} = ∣ - 3.91990 \dots - (- 4.34658 \dots) ∣ = 0.42667 \dots

Δ v_{11} = 0.42667 \dots

v_{12} = - 3.53572 \dots : Δ v_{12} = 0.38418 \dots

f (v_{11}) = (- 3.91990 \dots)^{10} + (- 3.91990 \dots)^{9} + (- 3.91990 \dots)^{8} + (- 3.91990 \dots)^{7} + (- 3.91990 \dots)^{6} + (- 3.91990 \dots)^{5} + (- 3.91990 \dots)^{4} + (- 3.91990 \dots)^{3} + (- 3.91990 \dots)^{2} + (- 3.91990 \dots) + 2 = 682454.88750 \dots

f^{^{'}} (v_{11}) = 10 (- 3.91990 \dots)^{9} + 9 (- 3.91990 \dots)^{8} + 8 (- 3.91990 \dots)^{7} + 7 (- 3.91990 \dots)^{6} + 6 (- 3.91990 \dots)^{5} + 5 (- 3.91990 \dots)^{4} + 4 (- 3.91990 \dots)^{3} + 3 (- 3.91990 \dots)^{2} + 2 (- 3.91990 \dots) + 1 = - 1776382.62957 \dots

v_{12} = - 3.53572 \dots

Δ v_{12} = ∣ - 3.53572 \dots - (- 3.91990 \dots) ∣ = 0.38418 \dots

Δ v_{12} = 0.38418 \dots

v_{13} = - 3.18977 \dots : Δ v_{13} = 0.34594 \dots

f (v_{12}) = (- 3.53572 \dots)^{10} + (- 3.53572 \dots)^{9} + (- 3.53572 \dots)^{8} + (- 3.53572 \dots)^{7} + (- 3.53572 \dots)^{6} + (- 3.53572 \dots)^{5} + (- 3.53572 \dots)^{4} + (- 3.53572 \dots)^{3} + (- 3.53572 \dots)^{2} + (- 3.53572 \dots) + 2 = 238020.64656 \dots

f^{^{'}} (v_{12}) = 10 (- 3.53572 \dots)^{9} + 9 (- 3.53572 \dots)^{8} + 8 (- 3.53572 \dots)^{7} + 7 (- 3.53572 \dots)^{6} + 6 (- 3.53572 \dots)^{5} + 5 (- 3.53572 \dots)^{4} + 4 (- 3.53572 \dots)^{3} + 3 (- 3.53572 \dots)^{2} + 2 (- 3.53572 \dots) + 1 = - 688026.64330 \dots

v_{13} = - 3.18977 \dots

Δ v_{13} = ∣ - 3.18977 \dots - (- 3.53572 \dots) ∣ = 0.34594 \dots

Δ v_{13} = 0.34594 \dots

v_{14} = - 2.87822 \dots : Δ v_{14} = 0.31154 \dots

f (v_{13}) = (- 3.18977 \dots)^{10} + (- 3.18977 \dots)^{9} + (- 3.18977 \dots)^{8} + (- 3.18977 \dots)^{7} + (- 3.18977 \dots)^{6} + (- 3.18977 \dots)^{5} + (- 3.18977 \dots)^{4} + (- 3.18977 \dots)^{3} + (- 3.18977 \dots)^{2} + (- 3.18977 \dots) + 2 = 83018.73554 \dots

f^{^{'}} (v_{13}) = 10 (- 3.18977 \dots)^{9} + 9 (- 3.18977 \dots)^{8} + 8 (- 3.18977 \dots)^{7} + 7 (- 3.18977 \dots)^{6} + 6 (- 3.18977 \dots)^{5} + 5 (- 3.18977 \dots)^{4} + 4 (- 3.18977 \dots)^{3} + 3 (- 3.18977 \dots)^{2} + 2 (- 3.18977 \dots) + 1 = - 266473.07023 \dots

v_{14} = - 2.87822 \dots

Δ v_{14} = ∣ - 2.87822 \dots - (- 3.18977 \dots) ∣ = 0.31154 \dots

Δ v_{14} = 0.31154 \dots

v_{15} = - 2.59762 \dots : Δ v_{15} = 0.28060 \dots

f (v_{14}) = (- 2.87822 \dots)^{10} + (- 2.87822 \dots)^{9} + (- 2.87822 \dots)^{8} + (- 2.87822 \dots)^{7} + (- 2.87822 \dots)^{6} + (- 2.87822 \dots)^{5} + (- 2.87822 \dots)^{4} + (- 2.87822 \dots)^{3} + (- 2.87822 \dots)^{2} + (- 2.87822 \dots) + 2 = 28957.64859 \dots

f^{^{'}} (v_{14}) = 10 (- 2.87822 \dots)^{9} + 9 (- 2.87822 \dots)^{8} + 8 (- 2.87822 \dots)^{7} + 7 (- 2.87822 \dots)^{6} + 6 (- 2.87822 \dots)^{5} + 5 (- 2.87822 \dots)^{4} + 4 (- 2.87822 \dots)^{3} + 3 (- 2.87822 \dots)^{2} + 2 (- 2.87822 \dots) + 1 = - 103198.93705 \dots

v_{15} = - 2.59762 \dots

Δ v_{15} = ∣ - 2.59762 \dots - (- 2.87822 \dots) ∣ = 0.28060 \dots

Δ v_{15} = 0.28060 \dots

v_{16} = - 2.34485 \dots : Δ v_{16} = 0.25277 \dots

f (v_{15}) = (- 2.59762 \dots)^{10} + (- 2.59762 \dots)^{9} + (- 2.59762 \dots)^{8} + (- 2.59762 \dots)^{7} + (- 2.59762 \dots)^{6} + (- 2.59762 \dots)^{5} + (- 2.59762 \dots)^{4} + (- 2.59762 \dots)^{3} + (- 2.59762 \dots)^{2} + (- 2.59762 \dots) + 2 = 10101.49087 \dots

f^{^{'}} (v_{15}) = 10 (- 2.59762 \dots)^{9} + 9 (- 2.59762 \dots)^{8} + 8 (- 2.59762 \dots)^{7} + 7 (- 2.59762 \dots)^{6} + 6 (- 2.59762 \dots)^{5} + 5 (- 2.59762 \dots)^{4} + 4 (- 2.59762 \dots)^{3} + 3 (- 2.59762 \dots)^{2} + 2 (- 2.59762 \dots) + 1 = - 39963.14592 \dots

v_{16} = - 2.34485 \dots

Δ v_{16} = ∣ - 2.34485 \dots - (- 2.59762 \dots) ∣ = 0.25277 \dots

Δ v_{16} = 0.25277 \dots

v_{17} = - 2.11709 \dots : Δ v_{17} = 0.22776 \dots

f (v_{16}) = (- 2.34485 \dots)^{10} + (- 2.34485 \dots)^{9} + (- 2.34485 \dots)^{8} + (- 2.34485 \dots)^{7} + (- 2.34485 \dots)^{6} + (- 2.34485 \dots)^{5} + (- 2.34485 \dots)^{4} + (- 2.34485 \dots)^{3} + (- 2.34485 \dots)^{2} + (- 2.34485 \dots) + 2 = 3524.23142 \dots

f^{^{'}} (v_{16}) = 10 (- 2.34485 \dots)^{9} + 9 (- 2.34485 \dots)^{8} + 8 (- 2.34485 \dots)^{7} + 7 (- 2.34485 \dots)^{6} + 6 (- 2.34485 \dots)^{5} + 5 (- 2.34485 \dots)^{4} + 4 (- 2.34485 \dots)^{3} + 3 (- 2.34485 \dots)^{2} + 2 (- 2.34485 \dots) + 1 = - 15473.15593 \dots

v_{17} = - 2.11709 \dots

Δ v_{17} = ∣ - 2.11709 \dots - (- 2.34485 \dots) ∣ = 0.22776 \dots

Δ v_{17} = 0.22776 \dots

v_{18} = - 1.91174 \dots : Δ v_{18} = 0.20535 \dots

f (v_{17}) = (- 2.11709 \dots)^{10} + (- 2.11709 \dots)^{9} + (- 2.11709 \dots)^{8} + (- 2.11709 \dots)^{7} + (- 2.11709 \dots)^{6} + (- 2.11709 \dots)^{5} + (- 2.11709 \dots)^{4} + (- 2.11709 \dots)^{3} + (- 2.11709 \dots)^{2} + (- 2.11709 \dots) + 2 = 1229.86602 \dots

f^{^{'}} (v_{17}) = 10 (- 2.11709 \dots)^{9} + 9 (- 2.11709 \dots)^{8} + 8 (- 2.11709 \dots)^{7} + 7 (- 2.11709 \dots)^{6} + 6 (- 2.11709 \dots)^{5} + 5 (- 2.11709 \dots)^{4} + 4 (- 2.11709 \dots)^{3} + 3 (- 2.11709 \dots)^{2} + 2 (- 2.11709 \dots) + 1 = - 5989.04315 \dots

v_{18} = - 1.91174 \dots

Δ v_{18} = ∣ - 1.91174 \dots - (- 2.11709 \dots) ∣ = 0.20535 \dots

Δ v_{18} = 0.20535 \dots

v_{19} = - 1.72632 \dots : Δ v_{19} = 0.18541 \dots

f (v_{18}) = (- 1.91174 \dots)^{10} + (- 1.91174 \dots)^{9} + (- 1.91174 \dots)^{8} + (- 1.91174 \dots)^{7} + (- 1.91174 \dots)^{6} + (- 1.91174 \dots)^{5} + (- 1.91174 \dots)^{4} + (- 1.91174 \dots)^{3} + (- 1.91174 \dots)^{2} + (- 1.91174 \dots) + 2 = 429.46495 \dots

f^{^{'}} (v_{18}) = 10 (- 1.91174 \dots)^{9} + 9 (- 1.91174 \dots)^{8} + 8 (- 1.91174 \dots)^{7} + 7 (- 1.91174 \dots)^{6} + 6 (- 1.91174 \dots)^{5} + 5 (- 1.91174 \dots)^{4} + 4 (- 1.91174 \dots)^{3} + 3 (- 1.91174 \dots)^{2} + 2 (- 1.91174 \dots) + 1 = - 2316.22500 \dots

v_{19} = - 1.72632 \dots

Δ v_{19} = ∣ - 1.72632 \dots - (- 1.91174 \dots) ∣ = 0.18541 \dots

Δ v_{19} = 0.18541 \dots

v_{20} = - 1.55824 \dots : Δ v_{20} = 0.16807 \dots

f (v_{19}) = (- 1.72632 \dots)^{10} + (- 1.72632 \dots)^{9} + (- 1.72632 \dots)^{8} + (- 1.72632 \dots)^{7} + (- 1.72632 \dots)^{6} + (- 1.72632 \dots)^{5} + (- 1.72632 \dots)^{4} + (- 1.72632 \dots)^{3} + (- 1.72632 \dots)^{2} + (- 1.72632 \dots) + 2 = 150.22439 \dots

f^{^{'}} (v_{19}) = 10 (- 1.72632 \dots)^{9} + 9 (- 1.72632 \dots)^{8} + 8 (- 1.72632 \dots)^{7} + 7 (- 1.72632 \dots)^{6} + 6 (- 1.72632 \dots)^{5} + 5 (- 1.72632 \dots)^{4} + 4 (- 1.72632 \dots)^{3} + 3 (- 1.72632 \dots)^{2} + 2 (- 1.72632 \dots) + 1 = - 893.77355 \dots

v_{20} = - 1.55824 \dots

Δ v_{20} = ∣ - 1.55824 \dots - (- 1.72632 \dots) ∣ = 0.16807 \dots

Δ v_{20} = 0.16807 \dots

v_{21} = - 1.40415 \dots : Δ v_{21} = 0.15408 \dots

f (v_{20}) = (- 1.55824 \dots)^{10} + (- 1.55824 \dots)^{9} + (- 1.55824 \dots)^{8} + (- 1.55824 \dots)^{7} + (- 1.55824 \dots)^{6} + (- 1.55824 \dots)^{5} + (- 1.55824 \dots)^{4} + (- 1.55824 \dots)^{3} + (- 1.55824 \dots)^{2} + (- 1.55824 \dots) + 2 = 52.80177 \dots

f^{^{'}} (v_{20}) = 10 (- 1.55824 \dots)^{9} + 9 (- 1.55824 \dots)^{8} + 8 (- 1.55824 \dots)^{7} + 7 (- 1.55824 \dots)^{6} + 6 (- 1.55824 \dots)^{5} + 5 (- 1.55824 \dots)^{4} + 4 (- 1.55824 \dots)^{3} + 3 (- 1.55824 \dots)^{2} + 2 (- 1.55824 \dots) + 1 = - 342.67162 \dots

v_{21} = - 1.40415 \dots

Δ v_{21} = ∣ - 1.40415 \dots - (- 1.55824 \dots) ∣ = 0.15408 \dots

Δ v_{21} = 0.15408 \dots

v_{22} = - 1.25818 \dots : Δ v_{22} = 0.14597 \dots

f (v_{21}) = (- 1.40415 \dots)^{10} + (- 1.40415 \dots)^{9} + (- 1.40415 \dots)^{8} + (- 1.40415 \dots)^{7} + (- 1.40415 \dots)^{6} + (- 1.40415 \dots)^{5} + (- 1.40415 \dots)^{4} + (- 1.40415 \dots)^{3} + (- 1.40415 \dots)^{2} + (- 1.40415 \dots) + 2 = 18.81846 \dots

f^{^{'}} (v_{21}) = 10 (- 1.40415 \dots)^{9} + 9 (- 1.40415 \dots)^{8} + 8 (- 1.40415 \dots)^{7} + 7 (- 1.40415 \dots)^{6} + 6 (- 1.40415 \dots)^{5} + 5 (- 1.40415 \dots)^{4} + 4 (- 1.40415 \dots)^{3} + 3 (- 1.40415 \dots)^{2} + 2 (- 1.40415 \dots) + 1 = - 128.91771 \dots

v_{22} = - 1.25818 \dots

Δ v_{22} = ∣ - 1.25818 \dots - (- 1.40415 \dots) ∣ = 0.14597 \dots

Δ v_{22} = 0.14597 \dots

v_{23} = - 1.10566 \dots : Δ v_{23} = 0.15251 \dots

f (v_{22}) = (- 1.25818 \dots)^{10} + (- 1.25818 \dots)^{9} + (- 1.25818 \dots)^{8} + (- 1.25818 \dots)^{7} + (- 1.25818 \dots)^{6} + (- 1.25818 \dots)^{5} + (- 1.25818 \dots)^{4} + (- 1.25818 \dots)^{3} + (- 1.25818 \dots)^{2} + (- 1.25818 \dots) + 2 = 6.98182 \dots

f^{^{'}} (v_{22}) = 10 (- 1.25818 \dots)^{9} + 9 (- 1.25818 \dots)^{8} + 8 (- 1.25818 \dots)^{7} + 7 (- 1.25818 \dots)^{6} + 6 (- 1.25818 \dots)^{5} + 5 (- 1.25818 \dots)^{4} + 4 (- 1.25818 \dots)^{3} + 3 (- 1.25818 \dots)^{2} + 2 (- 1.25818 \dots) + 1 = - 45.77704 \dots

v_{23} = - 1.10566 \dots

Δ v_{23} = ∣ - 1.10566 \dots - (- 1.25818 \dots) ∣ = 0.15251 \dots

Δ v_{23} = 0.15251 \dots

Não se pode encontrar solução

A solução é

Sem solu \overset{c}{\c} \tilde{a} o para v \in R

A solução é

v = 1

v = 1

Substitua

v = u^{2},

solucione para

u

Resolver

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

Para

x^{2} = f (a)

as soluções são

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Aplicar a regra

1 = 1

= 1

- 1 = - 1

- 1

Aplicar a regra

1 = 1

= - 1

u = 1, u = - 1

As soluções são

u = 1, u = - 1

Substituir na equação

u = sec (x)

sec (x) = 1, sec (x) = - 1

sec (x) = 1, sec (x) = - 1

sec (x) = 1 : x = 2 πn

sec (x) = 1

Soluções gerais para

sec (x) = 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

Soluções gerais para

sec (x) = - 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

Combinar toda as soluções

x = 2 πn, x = π + 2 πn

Gráfico

Exemplos populares

2sec(x)=1+cos(x)

2 sec (x) = 1 + cos (x)

sin^2(x)+cos(x)-cos^2(x)=0

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

8(1-sin^2(x))+2sin(x)-7=0

8 (1 - sin^{2} (x)) + 2 sin (x) - 7 = 0

cot(5x)=1

cot (5 x) = 1

tan^2(x)-6tan(x)+1=0

tan^{2} (x) - 6 tan (x) + 1 = 0