Certainly! Let's solve this problem step-by-step.
We are given that [tex]\(\sin(u) = -\frac{7}{25}\)[/tex] and [tex]\(\cos(v) = -\frac{4}{5}\)[/tex], and both [tex]\(u\)[/tex] and [tex]\(v\)[/tex] are in quadrant III. In quadrant III, both sine and cosine functions are negative.
### Step 1: Finding [tex]\(\cos(u)\)[/tex]
To find [tex]\(\cos(u)\)[/tex], we use the Pythagorean identity:
[tex]\[
\sin^2(u) + \cos^2(u) = 1
\][/tex]
Given [tex]\(\sin(u) = -\frac{7}{25}\)[/tex]:
[tex]\[
\left( -\frac{7}{25} \right)^2 + \cos^2(u) = 1
\][/tex]
[tex]\[
\frac{49}{625} + \cos^2(u) = 1
\][/tex]
[tex]\[
\cos^2(u) = 1 - \frac{49}{625}
\][/tex]
[tex]\[
\cos^2(u) = \frac{625}{625} - \frac{49}{625}
\][/tex]
[tex]\[
\cos^2(u) = \frac{576}{625}
\][/tex]
Since [tex]\(u\)[/tex] is in the third quadrant, [tex]\(\cos(u)\)[/tex] is negative:
[tex]\[
\cos(u) = -\sqrt{\frac{576}{625}} = -\frac{24}{25}
\][/tex]
### Step 2: Finding [tex]\(\sin(v)\)[/tex]
We now use the Pythagorean identity to find [tex]\(\sin(v)\)[/tex]:
[tex]\[
\sin^2(v) + \cos^2(v) = 1
\][/tex]
Given [tex]\(\cos(v) = -\frac{4}{5}\)[/tex]:
[tex]\[
\sin^2(v) + \left( -\frac{4}{5} \right)^2 = 1
\][/tex]
[tex]\[
\sin^2(v) + \frac{16}{25} = 1
\][/tex]
[tex]\[
\sin^2(v) = 1 - \frac{16}{25}
\][/tex]
[tex]\[
\sin^2(v) = \frac{25}{25} - \frac{16}{25}
\][/tex]
[tex]\[
\sin^2(v) = \frac{9}{25}
\][/tex]
Since [tex]\(v\)[/tex] is in the third quadrant, [tex]\(\sin(v)\)[/tex] is negative:
[tex]\[
\sin(v) = -\sqrt{\frac{9}{25}} = -\frac{3}{5}
\][/tex]
### Step 3: Finding [tex]\(\cos(u - v)\)[/tex]
Use the cosine angle subtraction formula:
[tex]\[
\cos(u - v) = \cos(u) \cos(v) + \sin(u) \sin(v)
\][/tex]
Substitute the known values:
[tex]\[
\cos(u - v) = \left( -\frac{24}{25} \right) \left( -\frac{4}{5} \right) + \left( -\frac{7}{25} \right) \left( -\frac{3}{5} \right)
\][/tex]
Perform the multiplications:
[tex]\[
\cos(u - v) = \frac{96}{125} + \frac{21}{125}
\][/tex]
Add the results:
[tex]\[
\cos(u - v) = \frac{96 + 21}{125} = \frac{117}{125}
\][/tex]
Hence, the exact value of [tex]\(\cos(u - v)\)[/tex] is:
[tex]\[
\boxed{\frac{117}{125}}
\][/tex]
