Question

o ângulo agudo que as retas y=x-1 e y=3 determinam, possui uma tangente igual a:
a)raiz de 3 sobre 3
b)raiz de 3
c)raiz de 2 sobre 2
d)raiz de 2
e) 1​

Answer 1

✅ Após resolver os cálculos, concluímos que a tangente procurada da referida questão é:

                             $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf \tan\theta = 1\:\:\:}}\end{gathered} }$

Portanto, a opção correta é:

                       $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf Alternativa\:E\:\:\:}}\end{gathered} }$

Sejam as retas dadas:

                             $\Large\begin{cases} r: y = x - 1\\s: y = 3\end{cases}$

Para calcular a tangente do ângulo que as retas formam entre si devemos utilizar a seguinte fórmula:

$\Large\displaystyle\begin{gathered} \bf I\end{gathered}$         $\Large\displaystyle\text{ \begin{gathered} \tan\theta = \tan\left[\arctan\bigg(\bigg|\frac{m_{s} - m_{r}}{1 + m_{s}\cdot m_{r}}\bigg|\bigg)\right]\end{gathered} }$

Observe que para realizar os cálculos devemos recuperar os coeficientes angulares de ambas as retas dadas. Então, temos:

• Recuperando o coeficiente angular da reta "r":

            $\Large\displaystyle\begin{gathered} \textrm{Se}\:r:y = x - 1\Longrightarrow m_{r} = 1\end{gathered}$

                                $\Large\displaystyle\begin{gathered} \therefore\:\:m_{r} = 1\end{gathered}$

• Recuperando o coeficiente angular da reta "s":

                $\Large\displaystyle\begin{gathered} \textrm{Se}\:s:y = 3\Longrightarrow m_{s} = 0\end{gathered}$

                               $\Large\displaystyle\begin{gathered} \therefore\:\:m_{s} = 0\end{gathered}$

Substituindo os  valores dos coeficientes angulares das respectivas retas na equação "I", temos:

          $\Large\displaystyle\begin{gathered} \tan\theta = \tan\left[\arctan\bigg(\bigg|\frac{0 - 1}{1 + 0\cdot1}\bigg|\bigg)\right]\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = \tan\left[\arctan\bigg(\bigg|\frac{-1}{1}\bigg|\bigg)\right]\end{gathered}$

                       $\Large\displaystyle\begin{gathered} = \tan\left[\arctan(|-1|)\right]\end{gathered}$

                       $\Large\displaystyle\begin{gathered} = \tan\left[\arctan(1)\right]\end{gathered}$

                       $\Large\displaystyle\begin{gathered} = \tan\left[45^{\circ}\right]\end{gathered}$

                       $\Large\displaystyle\begin{gathered} = 1\end{gathered}$

✅ Portanto, a tangente do ângulo procurado é:

           $\Large\displaystyle\begin{gathered} \tan\theta = 1\end{gathered}$  

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

Saiba mais:

1. https://brainly.com.br/tarefa/227145
2. https://brainly.com.br/tarefa/52000661
3. https://brainly.com.br/tarefa/53671929

$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$