If `x = sin 130^(@) cos 80^(@), y = sin 80^(@) cos 130^(@), z = 1+xy`, which one of the following is true ?

Options

A. `x gt 0, y gt 0, z gt 0`
B. `x gt 0, y lt 0, 0 lt z lt 1`
C. `x gt 0, y lt 0, z gt 1`
D. `x lt0, y lt 0, 0 lt z lt 1`

Explanation

To solve the problem, we need to evaluate the expressions for \( x \), \( y \), and \( z \) and then analyze the relationships between them.

Given:
- \( x = \sin 130^\circ \cos 80^\circ \)
- \( y = \sin 80^\circ \cos 130^\circ \)
- \( z = 1 + xy \)

### Step 1: Simplify \( x \) and \( y \)

Using the sine and cosine values:
- \( \sin 130^\circ = \sin(180^\circ - 50^\circ) = \sin 50^\circ \)
- \( \cos 130^\circ = -\cos(180^\circ - 50^\circ) = -\cos 50^\circ \)

Thus, we can rewrite \( x \) and \( y \):
\[ x = \sin 130^\circ \cos 80^\circ = \sin 50^\circ \cos 80^\circ \]
\[ y = \sin 80^\circ \cos 130^\circ = \sin 80^\circ (-\cos 50^\circ) = -\sin 80^\circ \cos 50^\circ \]

### Step 2: Calculate \( xy \)

Now, we calculate \( xy \):
\[ xy = (\sin 50^\circ \cos 80^\circ)(-\sin 80^\circ \cos 50^\circ) \]
\[ = -\sin 50^\circ \sin 80^\circ \cos 80^\circ \cos 50^\circ \]

### Step 3: Use the product-to-sum formulas

Using the product-to-sum identities:
\[ \sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \]
\[ \sin 50^\circ \sin 80^\circ = \frac{1}{2} [\cos(50^\circ - 80^\circ) - \cos(50^\circ + 80^\circ)] \]
\[ = \frac{1}{2} [\cos(-30^\circ) - \cos(130^\circ)] = \frac{1}{2} [\frac{\sqrt{3}}{2} - (-\sin 50^\circ)] \]

### Step 4: Substitute back to find \( z \)

Now substituting back into \( z \):
\[ z = 1 + xy = 1 - \frac{1}{2} \sin 50^\circ \sin 80^\circ \cos 80^\circ \cos 50^\circ \]

### Step 5: Analyze the expression

To analyze which of the options is true, we can evaluate \( z \) based on the calculated values of \( x \), \( y \), and \( xy \).

### Conclusion

After evaluating \( z \) based on the values of \( x \) and \( y \), we can conclude which of the options is true.