To find [tex]\(\tan(\theta)\)[/tex] using the given values for [tex]\(\sin(\theta)\)[/tex] and [tex]\(\cos(\theta)\)[/tex], we can use the quotient identity for tangent. The quotient identity states that:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
Given:
[tex]\[
\sin(\theta) = -\frac{\sqrt{7}}{6}
\][/tex]
[tex]\[
\cos(\theta) = -\frac{5}{6}
\][/tex]
We substitute these values into the quotient identity:
[tex]\[
\tan(\theta) = \frac{-\frac{\sqrt{7}}{6}}{-\frac{5}{6}}
\][/tex]
When dividing fractions, the division of [tex]\(\frac{a}{b}\)[/tex] by [tex]\(\frac{c}{d}\)[/tex] can be performed by multiplying [tex]\(a\)[/tex] by the reciprocal of [tex]\(b\)[/tex]:
[tex]\[
\tan(\theta) = -\frac{\sqrt{7}}{6} \div -\frac{5}{6} = -\frac{\sqrt{7}}{6} \times -\frac{6}{5}
\][/tex]
Now, multiply the numerators together and the denominators together:
[tex]\[
\tan(\theta) = \frac{(-\sqrt{7}) \times (-6)}{6 \times 5} = \frac{6\sqrt{7}}{30}
\][/tex]
Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 6:
[tex]\[
\tan(\theta) = \frac{\sqrt{7}}{5}
\][/tex]
Therefore, [tex]\(\tan(\theta) = \frac{\sqrt{7}}{5}\)[/tex].
In decimal form, this value is approximately equal to 0.529150262212918.
