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Use tables to find the acute angle $\theta $, given that:
$\tan \theta =3.1749$

Last updated date: 25th Jul 2024
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Hint: To find the acute angle $\theta $, we will look for the given value $3.1749$ or less than $3.1749$ in the table. If we are able to get the exact value $3.1749$ in the table, then it will be required to angle $\theta $. If we get the value less than $3.1749$, then we will subtract the obtained value less than $3.1749$ from $3.1749$ and we will look at the difference value in the table of mean difference. After combining this obtained value, we will get the acute angle of $\theta $.

Complete step by step answer:
Since, we already have the table of tangents. We will look in the table for the angle that has the value $3.1749$ or less than $3.1749$.
$\Rightarrow \text{Value of}\tan \theta \le 3.1749$
As we know that in the table of tangent that there are not any value of $\theta $ is equal to $3.1749$ but at the row of angle of $72{}^\circ $ and in the column of $30'$, we observe the value of $\theta $ as $3.1716$ that is less than $3.1749$.
$\Rightarrow \text{Value of}\tan 72{}^\circ 30'=3.1716<3.1749$
Now, we will find the difference of $3.1716$ and $3.1749$ as:
  & =3.1749-3.1716 \\
 & =0.0033 \\
Again, we will see the same row of angle but we will look for the column of mean difference in the table of tangents for the value of $0.0033$. We can find this value in a column of $1'$ of mean difference. So, we will add $1'$ in $72{}^\circ 30'$ to get the required angle as:
$\Rightarrow \text{Value of}\tan \left( 72{}^\circ 30'+1' \right)=3.1749$
After solving this, we will have:
$\Rightarrow \text{Value of}\tan \left( 72{}^\circ 31' \right)=3.1749$
And we already have $\tan \theta =3.1749$ from the question. So, we can write the above step as:
\[\Rightarrow \tan \left( 72{}^\circ 31' \right)=\tan \theta \]
After comparing both sides, we will get the value of $\theta $ as:
$\Rightarrow \theta =72{}^\circ 31'$
Hence, after using the table the acute angle $\theta $ is equal to $72{}^\circ 31'$.

Note: we can check the solution by looking for the value of \[\tan \left( 72{}^\circ 31' \right)\] in the table as:
As we have the angle of tangent as $72{}^\circ 31'$, we will observe the nearest value of $72{}^\circ 31'$ in the row of $72{}^\circ $ and we will get the nearest value in the column of $30'$ that is $0.1716$ as:
\[\Rightarrow \tan \left( 72{}^\circ 30' \right)=0.1716\]
Now, the remaining angle is $1'$. So, we will look for that value in the column of the table of mean difference in the row of $72{}^\circ $ and will get the value $0.0033$. Here, we will add it to get the required answer as:
\[\Rightarrow \tan \left( 72{}^\circ 30'+1' \right)=0.1716+0.0033\]
After addition, we will have:
\[\Rightarrow \tan \left( 72{}^\circ 31' \right)=0.1749\]
Since, we got the given value from the question. Hence, the solution is correct.