To find the length of $WY$ in $\Delta XYZ$, we'll use the given information and apply trigonometric principles. Here's the step-by-step solution:

Step 1: Analyze the given information

• $\angle Y = 90^\circ$
• $\angle X = 5^\circ$
• $\angle ZWY = 67^\circ$
• $XW = 570$ units

Step 2: Determine the angle $\angle Z$

Since $\angle Y = 90^\circ$ in $\Delta XYZ$, the third angle $\angle Z$ can be calculated as: $\angle Z = 180^\circ - \angle X - \angle Y = 180^\circ - 5^\circ - 90^\circ = 85^\circ$

Step 3: Apply the Law of Sines in $\Delta ZWY$

In $\Delta ZWY$, $\angle ZWY = 67^\circ$. We can find $\angle Z$ as follows: $\angle ZWY + \angle Z = 85^\circ$ So, $\angle ZWY = 67^\circ$ and $\angle WYZ = 85^\circ - 67^\circ = 18^\circ$.

Now, apply the Law of Sines in $\Delta ZWY$: $\frac{XW}{\sin(\angle WYZ)} = \frac{WY}{\sin(\angle X)}$ Substituting the values: $\frac{570}{\sin(18^\circ)} = \frac{WY}{\sin(5^\circ)}$

Step 4: Solve for $WY$

First, calculate $\sin(18^\circ)$ and $\sin(5^\circ)$ using a calculator: $\sin(18^\circ) \approx 0.3090$ $\sin(5^\circ) \approx 0.0872$

Now, substitute these values into the equation: $\frac{570}{0.3090} = \frac{WY}{0.0872}$ $WY = \frac{570 \times 0.0872}{0.3090} \approx 160.7$

Step 5: Round to the nearest integer

So, the length of $WY$ to the nearest integer is approximately $161$ units.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the solution change if $XW$ were a different length?
2. What is the significance of the Law of Sines in solving triangle problems?
3. How would you solve the problem if only the angles were given, and no side lengths?
4. What would be the approach if $\angle ZWY$ was obtuse instead of acute?
5. Can you generalize this approach for any right triangle with an additional angle?

Tip: Always double-check trigonometric values to ensure accuracy in angle-based calculations.