13. Given that $$ an (A-B)=4 $$ and that $$ an A=\frac{1}{3} $$ find the exact value of $$ an B $$.

Question

13. Given that $$ 	an (A-B)=4 $$ and that $$ 	an A=\frac{1}{3} $$ find the exact value of $$ 	an B $$.

UpStudy AI · Accepted Answer

To find the exact value of $$ \tan B $$ given that $$ \tan(A - B) = 4 $$ and $$ \tan A = \frac{1}{3} $$, we can use the tangent subtraction formula:

$$
\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
$$

Substituting the given values:

$$
4 = \frac{\frac{1}{3} - \tan B}{1 + \frac{1}{3} \tan B}
$$

Multiply both sides by the denominator to eliminate the fraction:

$$
4 \left(1 + \frac{1}{3} \tan B\right) = \frac{1}{3} - \tan B
$$

Expand the left side:

$$
4 + \frac{4}{3} \tan B = \frac{1}{3} - \tan B
$$

Bring all terms involving $$ \tan B $$ to one side and constant terms to the other:

$$
4 - \frac{1}{3} + \frac{4}{3} \tan B + \tan B = 0
$$

Combine like terms:

$$
\frac{11}{3} + \frac{7}{3} \tan B = 0
$$

Solving for $$ \tan B $$:

$$
\frac{7}{3} \tan B = -\frac{11}{3}
$$

$$
\tan B = -\frac{11}{7}
$$

**Answer:** $$ \tan B = -\dfrac{11}{7} $$
$$ \tan B = -\dfrac{11}{7} $$

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