To find the area of a parallelogram given two sides and the angle between them, you can use the formula:
[tex]\[ \text{Area} = a \times b \times \sin(\theta) \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the sides of the parallelogram, and [tex]\( \theta \)[/tex] is the angle between them.
Let's break down the steps:
1. Identify the sides and angle:
- [tex]\( a = 6 \)[/tex] feet (one side)
- [tex]\( b = 10 \)[/tex] feet (the other side)
- [tex]\( \theta = 45^\circ \)[/tex] (the angle between the sides)
2. Convert the angle to radians:
- Trigonometric functions in most calculations use angles in radians. The conversion formula from degrees to radians is:
[tex]\[ \text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right) \][/tex]
- Thus, converting [tex]\( 45^\circ \)[/tex] to radians:
[tex]\[ 45^\circ \times \left(\frac{\pi}{180}\right) = \frac{\pi}{4} \][/tex]
- This results in approximately [tex]\( 0.7854 \)[/tex] radians.
3. Calculate the sine of the angle:
- For [tex]\( \theta = \frac{\pi}{4} \)[/tex], or [tex]\( 45^\circ \)[/tex], the sine value is:
[tex]\[ \sin\left(\frac{\pi}{4}\right) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071 \][/tex]
4. Calculate the area:
- Substitute the values into the area formula:
[tex]\[ \text{Area} = 6 \text{ ft} \times 10 \text{ ft} \times 0.7071 \][/tex]
- This simplifies to:
[tex]\[ \text{Area} \approx 42.4264 \text{ ft}^2 \][/tex]
Therefore, the area of the parallelogram is approximately [tex]\( 42.43 \)[/tex] square feet.
