Find the exact value of the trigonometric expression given that sin u = - and cos v = - - 25 (Both u and v are in Quadrant III.) cos(u + v)
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![### Problem Statement
Find the exact value of the trigonometric expression given that \(\sin u = -\frac{7}{25}\) and \(\cos v = -\frac{4}{5}\). (Both \(u\) and \(v\) are in Quadrant III.)
#### Expression to Find:
\[
\cos(u + v)
\]
#### Textbox:
\[ \boxed{ \quad } \]
### Explanation
The problem given involves calculating the cosine of the sum of two angles, where both angles are located in the third quadrant of the trigonometric circle. Specifically, we know the sine of \( u \) and the cosine of \( v \).
**Steps to solve the problem:**
1. **Find \(\cos u\)** using the Pythagorean identity \(\sin^2 u + \cos^2 u = 1\):
\[
\cos u = -\sqrt{1 - \sin^2 u}
\]
Since \(u\) is in the third quadrant, where both sine and cosine are negative:
\[
\cos u = -\sqrt{1 - \left( -\frac{7}{25} \right)^2}
\]
2. **Find \(\sin v\)** using the Pythagorean identity:
\[
\sin v = -\sqrt{1 - \cos^2 v}
\]
Since \(v\) is in the third quadrant, where both sine and cosine are negative:
\[
\sin v = -\sqrt{1 - \left( -\frac{4}{5} \right)^2}
\]
3. **Find \(\cos(u + v)\)** using the sum formula for cosine:
\[
\cos(u + v) = \cos u \cos v - \sin u \sin v
\]
By determining all the necessary trigonometric values, you can then plug them into the formula to find the exact value of the expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe49cee-acc8-403b-8d12-20ec3514f583%2F5b8080da-90fd-4b54-af32-fbf20b41c01f%2F7xtoq6q_processed.png&w=3840&q=75)
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