Trigonometria Exemplos

\sin (θ) = \frac{\sqrt{7}}{4}

$\sin (θ) = \frac{\sqrt{7}}{4}$ ,

\cos (θ) = \frac{3}{4}

$\cos (θ) = \frac{3}{4}$

Etapa 1

Para encontrar o valor de

\tan (θ)

$\tan (θ)$ , use o fato de que

\tan (θ) = \frac{\sin (θ)}{\cos (θ)}

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)}$ e, então, substitua nos valores conhecidos.

\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\frac{\sqrt{7}}{4}}{\frac{3}{4}}

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\frac{\sqrt{7}}{4}}{\frac{3}{4}}$

Etapa 2

Simplifique o resultado.

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Etapa 2.1

Multiplique o numerador pelo inverso do denominador.

\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{7}}{4} \cdot \frac{4}{3}

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{7}}{4} \cdot \frac{4}{3}$

Etapa 2.2

Cancele o fator comum de

4

$4$ .

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Etapa 2.2.1

Cancele o fator comum.

\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{7}}{4} \cdot \frac{4}{3}

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{7}}{4} \cdot \frac{4}{3}$

Etapa 2.2.2

Reescreva a expressão.

\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \sqrt{7} (\frac{1}{3})

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \sqrt{7} (\frac{1}{3})$

\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \sqrt{7} (\frac{1}{3})

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \sqrt{7} (\frac{1}{3})$

Etapa 2.3

Combine

\sqrt{7}

$\sqrt{7}$ e

\frac{1}{3}

$\frac{1}{3}$ .

\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{7}}{3}

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{7}}{3}$

\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{7}}{3}

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{7}}{3}$

Etapa 3

Para encontrar o valor de

\cot (θ)

$\cot (θ)$ , use o fato de que

\frac{1}{\tan (θ)}

$\frac{1}{\tan (θ)}$ e, então, substitua nos valores conhecidos.

\cot (θ) = \frac{1}{\tan (θ)} = \frac{1}{\frac{\sqrt{7}}{3}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{1}{\frac{\sqrt{7}}{3}}$

Etapa 4

Simplifique o resultado.

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Etapa 4.1

Multiplique o numerador pelo inverso do denominador.

\cot (θ) = \frac{1}{\tan (θ)} = 1 (\frac{3}{\sqrt{7}})

$\cot (θ) = \frac{1}{\tan (θ)} = 1 (\frac{3}{\sqrt{7}})$

Etapa 4.2

Multiplique

\frac{3}{\sqrt{7}}

$\frac{3}{\sqrt{7}}$ por

1

$1$ .

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3}{\sqrt{7}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3}{\sqrt{7}}$

Etapa 4.3

Multiplique

\frac{3}{\sqrt{7}}

$\frac{3}{\sqrt{7}}$ por

\frac{\sqrt{7}}{\sqrt{7}}

$\frac{\sqrt{7}}{\sqrt{7}}$ .

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Etapa 4.4

Combine e simplifique o denominador.

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Etapa 4.4.1

Multiplique

\frac{3}{\sqrt{7}}

$\frac{3}{\sqrt{7}}$ por

\frac{\sqrt{7}}{\sqrt{7}}

$\frac{\sqrt{7}}{\sqrt{7}}$ .

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Etapa 4.4.2

Eleve

\sqrt{7}

$\sqrt{7}$ à potência de

1

$1$ .

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Etapa 4.4.3

Eleve

\sqrt{7}

$\sqrt{7}$ à potência de

1

$1$ .

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Etapa 4.4.4

Use a regra da multiplicação de potências

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Etapa 4.4.5

Some

1

$1$ e

1

$1$ .

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{{\sqrt{7}}^{2}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{{\sqrt{7}}^{2}}$

Etapa 4.4.6

Reescreva

{\sqrt{7}}^{2}

${\sqrt{7}}^{2}$ como

7

$7$ .

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Etapa 4.4.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever

\sqrt{7}

$\sqrt{7}$ como

7^{\frac{1}{2}}

$7^{\frac{1}{2}}$ .

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Etapa 4.4.6.2

Aplique a regra da multiplicação de potências e multiplique os expoentes,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Etapa 4.4.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ e

2

$2$ .

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Etapa 4.4.6.4

Cancele o fator comum de

2

$2$ .

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Etapa 4.4.6.4.1

Cancele o fator comum.

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Etapa 4.4.6.4.2

Reescreva a expressão.

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}$

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}$

Etapa 4.4.6.5

Avalie o expoente.

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}$

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}$

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}$

\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{3 \sqrt{7}}{7}$

Etapa 5

Para encontrar o valor de

\sec (θ)

$\sec (θ)$ , use o fato de que

\frac{1}{\cos (θ)}

$\frac{1}{\cos (θ)}$ e, então, substitua nos valores conhecidos.

\sec (θ) = \frac{1}{\cos (θ)} = \frac{1}{\frac{3}{4}}

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{1}{\frac{3}{4}}$

Etapa 6

Simplifique o resultado.

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Etapa 6.1

Multiplique o numerador pelo inverso do denominador.

\sec (θ) = \frac{1}{\cos (θ)} = 1 (\frac{4}{3})

$\sec (θ) = \frac{1}{\cos (θ)} = 1 (\frac{4}{3})$

Etapa 6.2

Multiplique

\frac{4}{3}

$\frac{4}{3}$ por

1

$1$ .

\sec (θ) = \frac{1}{\cos (θ)} = \frac{4}{3}

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4}{3}$

\sec (θ) = \frac{1}{\cos (θ)} = \frac{4}{3}

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4}{3}$

Etapa 7

Para encontrar o valor de

\csc (θ)

$\csc (θ)$ , use o fato de que

\frac{1}{\sin (θ)}

$\frac{1}{\sin (θ)}$ e, então, substitua nos valores conhecidos.

\csc (θ) = \frac{1}{\sin (θ)} = \frac{1}{\frac{\sqrt{7}}{4}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{1}{\frac{\sqrt{7}}{4}}$

Etapa 8

Simplifique o resultado.

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Etapa 8.1

Multiplique o numerador pelo inverso do denominador.

\csc (θ) = \frac{1}{\sin (θ)} = 1 (\frac{4}{\sqrt{7}})

$\csc (θ) = \frac{1}{\sin (θ)} = 1 (\frac{4}{\sqrt{7}})$

Etapa 8.2

Multiplique

\frac{4}{\sqrt{7}}

$\frac{4}{\sqrt{7}}$ por

1

$1$ .

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4}{\sqrt{7}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4}{\sqrt{7}}$

Etapa 8.3

Multiplique

\frac{4}{\sqrt{7}}

$\frac{4}{\sqrt{7}}$ por

\frac{\sqrt{7}}{\sqrt{7}}

$\frac{\sqrt{7}}{\sqrt{7}}$ .

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Etapa 8.4

Combine e simplifique o denominador.

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Etapa 8.4.1

Multiplique

\frac{4}{\sqrt{7}}

$\frac{4}{\sqrt{7}}$ por

\frac{\sqrt{7}}{\sqrt{7}}

$\frac{\sqrt{7}}{\sqrt{7}}$ .

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Etapa 8.4.2

Eleve

\sqrt{7}

$\sqrt{7}$ à potência de

1

$1$ .

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Etapa 8.4.3

Eleve

\sqrt{7}

$\sqrt{7}$ à potência de

1

$1$ .

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Etapa 8.4.4

Use a regra da multiplicação de potências

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Etapa 8.4.5

Some

1

$1$ e

1

$1$ .

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{{\sqrt{7}}^{2}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{{\sqrt{7}}^{2}}$

Etapa 8.4.6

Reescreva

{\sqrt{7}}^{2}

${\sqrt{7}}^{2}$ como

7

$7$ .

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Etapa 8.4.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever

\sqrt{7}

$\sqrt{7}$ como

7^{\frac{1}{2}}

$7^{\frac{1}{2}}$ .

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Etapa 8.4.6.2

Aplique a regra da multiplicação de potências e multiplique os expoentes,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Etapa 8.4.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ e

2

$2$ .

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Etapa 8.4.6.4

Cancele o fator comum de

2

$2$ .

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Etapa 8.4.6.4.1

Cancele o fator comum.

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Etapa 8.4.6.4.2

Reescreva a expressão.

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}$

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}$

Etapa 8.4.6.5

Avalie o expoente.

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}$

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}$

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}$

\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{4 \sqrt{7}}{7}$

Etapa 9

As funções trigonométricas encontradas são as seguintes:

\sin (θ) = \frac{\sqrt{7}}{4}

$\sin (θ) = \frac{\sqrt{7}}{4}$

\cos (θ) = \frac{3}{4}

$\cos (θ) = \frac{3}{4}$

\tan (θ) = \frac{\sqrt{7}}{3}

$\tan (θ) = \frac{\sqrt{7}}{3}$

\cot (θ) = \frac{3 \sqrt{7}}{7}

$\cot (θ) = \frac{3 \sqrt{7}}{7}$

\sec (θ) = \frac{4}{3}

$\sec (θ) = \frac{4}{3}$

\csc (θ) = \frac{4 \sqrt{7}}{7}

$\csc (θ) = \frac{4 \sqrt{7}}{7}$