In triangle XYZ, angle X measures 45 degrees, side y is 8 units, and side z is 6 units. Use the Law of Cosines to find the measure of angle Y.

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To find the measure of angle $$ Y $$ in triangle $$ XYZ $$ using the Law of Cosines, let's follow these steps:

### Given:
- Angle $$ X = 45^\circ $$
- Side $$ y = 8 $$ units (opposite angle $$ Y $$)
- Side $$ z = 6 $$ units (opposite angle $$ Z $$)

### Step 1: Apply the Law of Cosines to Find Side $$ x $$

The Law of Cosines states:
$$
x^2 = y^2 + z^2 - 2yz \cos(X)
$$

Plugging in the known values:
$$
x^2 = 8^2 + 6^2 - 2 \times 8 \times 6 \times \cos(45^\circ)
$$
$$
x^2 = 64 + 36 - 96 \times \frac{\sqrt{2}}{2}
$$
$$
x^2 = 100 - 48\sqrt{2}
$$
$$
x = \sqrt{100 - 48\sqrt{2}} \approx 5.67 \text{ units}
$$

### Step 2: Use the Law of Cosines to Find Angle $$ Y $$

Now, apply the Law of Cosines to angle $$ Y $$:
$$
y^2 = x^2 + z^2 - 2xz \cos(Y)
$$

Rearranging for $$ \cos(Y) $$:
$$
\cos(Y) = \frac{x^2 + z^2 - y^2}{2xz}
$$
$$
\cos(Y) = \frac{(100 - 48\sqrt{2}) + 36 - 64}{2 \times \sqrt{100 - 48\sqrt{2}} \times 6}
$$
$$
\cos(Y) = \frac{72 - 48\sqrt{2}}{12 \sqrt{100 - 48\sqrt{2}}}
$$
$$
\cos(Y) \approx \frac{0.343}{5.67} \approx 0.0605
$$

### Step 3: Calculate Angle $$ Y $$

Using the inverse cosine function:
$$
Y = \cos^{-1}(0.0605) \approx 86.5^\circ
$$

### Final Answer:
The measure of angle $$ Y $$ is approximately **86.5 degrees**.
Angle $$ Y $$ is approximately 86.5 degrees.

In triangle XYZ, angle X measures 45 degrees, side y is 8 units, and side z is 6 units. Use the Law of Cosines to find the measure of angle Y.

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