In ΔWXY, *w* = 670 cm, *y* = 800 cm and ∠Y=84°. Find all possible values of ∠W, *to the nearest 10th of a degree*.

In triangle \( \Delta WXY \), we are given:

- Side \( w = 670 \) cm (opposite angle \( W \)),
- Side \( y = 800 \) cm (opposite angle \( Y \)),
- \( \angle Y = 84^\circ \).

We are tasked with finding \( \angle W \) to the nearest tenth of a degree.

### Step 1: Use the Law of Sines
The Law of Sines states:
\[
\frac{w}{\sin W} = \frac{y}{\sin Y}
\]
Substitute the known values into the equation:
\[
\frac{670}{\sin W} = \frac{800}{\sin 84^\circ}
\]
Since \( \sin 84^\circ \approx 0.9945 \), the equation becomes:
\[
\frac{670}{\sin W} = \frac{800}{0.9945}
\]
\[
\frac{670}{\sin W} = 804.41
\]

### Step 2: Solve for \( \sin W \)
\[
\sin W = \frac{670}{804.41}
\]
\[
\sin W \approx 0.833
\]

### Step 3: Find \( W \)
Using the inverse sine function:
\[
W = \sin^{-1}(0.833)
\]
\[
W \approx 56.3^\circ
\]

### Step 4: Check for Ambiguous Case
Since \( \sin W = 0.833 \), there is a possibility of another solution due to the ambiguity in the sine function. The second possible value for \( W \) is:
\[
W_2 = 180^\circ - 56.3^\circ = 123.7^\circ
\]

### Step 5: Check if \( W_2 \) is valid
For \( W_2 \approx 123.7^\circ \), the sum of angles in the triangle would be:
\[
W_2 + \angle Y = 123.7^\circ + 84^\circ = 207.7^\circ
\]
Since this exceeds \( 180^\circ \), \( W_2 \) is not valid.

Thus, the only valid solution is \( W \approx 56.3^\circ \).

### Final Answer:
The value of \( \angle W \) is approximately **56.3°**.

---

Would you like more details or have any questions? Here are five related questions you could explore:
1. How can you find the third angle in the triangle after knowing \( W \) and \( Y \)?
2. What is the Law of Cosines, and how does it apply here?
3. How can the Law of Sines be used to find side \( x \) in this triangle?
4. What would happen if the angle given was obtuse? How would it affect the results?
5. Can you use trigonometric identities to verify the solution?

**Tip:** Always check for the ambiguous case when using the Law of Sines for non-right triangles.

In ΔWXY, w = 670 cm, y = 800 cm and ∠Y=84°. Find all possible values of ∠W, to the nearest 10th of a degree.

Explore how to find the angle ∠W in triangle ΔWXY, where w = 670 cm, y = 800 cm, and ∠Y = 84°, using the Law of Sines. This trigonometry problem includes a step-by-step breakdown of solving for ∠W, including checking for the ambiguous case.

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