In ΔUVW, w = 82 cm, u = 64 cm and ∠V=8°. Find the length of v, to the nearest centimeter.

We are given a triangle \( \triangle UVW \) with the following information:

- \( w = 82 \) cm (side opposite angle \( W \)),
- \( u = 64 \) cm (side opposite angle \( U \)),
- \( \angle V = 8^\circ \) (angle opposite side \( v \)).

We are to find the length of \( v \) (the side opposite \( \angle V \)) using the **Law of Cosines**. The Law of Cosines is given by:

\[
v^2 = u^2 + w^2 - 2uw \cdot \cos(\angle V)
\]

### Step-by-step solution:

1. First, substitute the known values into the formula:

\[
v^2 = 64^2 + 82^2 - 2(64)(82) \cdot \cos(8^\circ)
\]

2. Calculate the squares of the sides:

\[
v^2 = 4096 + 6724 - 2(64)(82) \cdot \cos(8^\circ)
\]

3. Simplify further:

\[
v^2 = 10820 - 10496 \cdot \cos(8^\circ)
\]

4. Now, find \( \cos(8^\circ) \). Using a calculator:

\[
\cos(8^\circ) \approx 0.9903
\]

5. Substitute \( \cos(8^\circ) \) into the equation:

\[
v^2 = 10820 - 10496 \cdot 0.9903
\]
\[
v^2 = 10820 - 10395.1
\]
\[
v^2 \approx 424.9
\]

6. Finally, take the square root of both sides:

\[
v \approx \sqrt{424.9} \approx 20.61
\]

To the nearest centimeter, the length of side \( v \) is approximately \( 21 \) cm.

---

Would you like more details or have any questions about the solution? Here are some related questions you might want to explore:

1. How does the Law of Cosines apply to obtuse triangles?
2. How can you derive the Law of Cosines from the Pythagorean theorem?
3. What would change if \( \angle V \) were larger, say 30 degrees?
4. How can we find the other angles of the triangle using the given information?
5. What is the Law of Sines, and how can it be used for this problem?

**Tip:** When using the Law of Cosines, always double-check the angle mode (degrees or radians) on your calculator!

Solve for the length of side v in triangle ΔUVW with given sides u = 64 cm, w = 82 cm, and angle ∠V = 8°. This problem involves using the Law of Cosines and is suitable for high school students studying trigonometry. The final result gives v ≈ 21 cm.

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