To solve this trigonometric problem, follow these steps:
1. Understand the Quadrant:
Given \(\cos(A) = -\frac{7}{9}\) and \(\sin(A) < 0\), angle \(A\) is in the third quadrant. In the third quadrant, both \(\cos\) and \(\sin\) are negative, and \(\tan\) is positive.
2. Find \(\sin(A)\):
We use the Pythagorean identity for sine and cosine:
[tex]\[
\sin^2(A) + \cos^2(A) = 1
\][/tex]
Substitute \(\cos(A) = -\frac{7}{9}\) into the equation:
[tex]\[
\sin^2(A) + \left(-\frac{7}{9}\right)^2 = 1
\][/tex]
[tex]\[
\sin^2(A) + \left(\frac{49}{81}\right) = 1
\][/tex]
Solve for \(\sin^2(A)\):
[tex]\[
\sin^2(A) = 1 - \frac{49}{81}
\][/tex]
[tex]\[
\sin^2(A) = \frac{81}{81} - \frac{49}{81}
\][/tex]
[tex]\[
\sin^2(A) = \frac{32}{81}
\][/tex]
Take the square root and select the negative value (since \(\sin(A)\) is negative in the third quadrant):
[tex]\[
\sin(A) = -\sqrt{\frac{32}{81}}
\][/tex]
Simplify the square root:
[tex]\[
\sin(A) = -\frac{\sqrt{32}}{9}
\][/tex]
[tex]\[
\sin(A) = -\frac{4\sqrt{2}}{9}
\][/tex]
3. Find \(\tan(A)\):
The tangent function is the ratio of the sine and cosine functions:
[tex]\[
\tan(A) = \frac{\sin(A)}{\cos(A)}
\][/tex]
Substitute \(\sin(A) = -\frac{4\sqrt{2}}{9}\) and \(\cos(A) = -\frac{7}{9}\):
[tex]\[
\tan(A) = \frac{-\frac{4\sqrt{2}}{9}}{-\frac{7}{9}}
\][/tex]
Simplify the expression:
[tex]\[
\tan(A) = \frac{4\sqrt{2}}{7}
\][/tex]
Now let's match the computed values with the given options:
- \(\sin(A) = -\frac{4\sqrt{2}}{9}\)
- \(\tan(A) = \frac{4\sqrt{2}}{7}\)
The answer matches the second choice:
[tex]\[
\sin (A)=-\frac{4 \sqrt{2}}{9}, \tan (A)=\frac{4 \sqrt{2}}{7}
\][/tex]
