To find the area of a parallelogram with given sides and an angle, we use the formula for the area of a parallelogram. The formula to calculate the area (A) is:
[tex]\[ A = b \cdot h \cdot \sin(\theta) \][/tex]
where:
- [tex]\( b \)[/tex] is the base length of the parallelogram.
- [tex]\( h \)[/tex] is the vertical height of the parallelogram relative to the base.
- [tex]\( \theta \)[/tex] is the angle between the base and the side (measured in degrees).
Given the problem:
- The base ([tex]\( b \)[/tex]) is 12 inches.
- The vertical height ([tex]\( h \)[/tex]) is 8 inches.
- The angle ([tex]\( \theta \)[/tex]) between the base and the adjacent side is 96 degrees.
Step-by-step solution:
1. Convert the angle from degrees to radians:
The angle given is in degrees, but we need it in radians for trigonometric calculations. The conversion factor from degrees to radians is:
[tex]\[
\theta \text{ (in radians)} = \theta \text{ (in degrees)} \times \frac{\pi}{180}
\][/tex]
So,
[tex]\[
96^\circ \times \frac{\pi}{180} = \text{ angle in radians (we can denote this as, for example, } \alpha \text{)}
\][/tex]
2. Compute the sine of the angle:
Using the converted angle in radians, we compute the sine of that angle:
[tex]\(\sin(\alpha)\)[/tex]
3. Substitute the values into the area formula:
[tex]\[
A = 12 \times 8 \times \sin(96^\circ)
\][/tex]
We already know the precise numerical result of this calculation. Hence, the area of the parallelogram is:
[tex]\[ \boxed{95.47410195535423 \text{ square inches}} \][/tex]
Thus, after performing these steps correctly, we determine the area of the parallelogram to be approximately 95.474 square inches.
