To find the measure of the included angle in a triangle given its area and two sides, we can use the formula for the area of a triangle when it involves trigonometric functions:
[tex]\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \][/tex]
Here,
- [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the two sides,
- [tex]\(C\)[/tex] is the included angle,
- and the area of the triangle is [tex]\(1256\)[/tex].
Given:
- [tex]\(\text{Area} = 1256\)[/tex]
- [tex]\(a = 51\)[/tex]
- [tex]\(b = 89\)[/tex]
We need to find the measure of angle [tex]\(C\)[/tex].
### Step-by-Step Solution:
1. Set up the formula for the area:
[tex]\[ 1256 = \frac{1}{2} \times 51 \times 89 \times \sin(C) \][/tex]
2. Solve for [tex]\(\sin(C)\)[/tex]:
Isolate [tex]\(\sin(C)\)[/tex]:
[tex]\[ \sin(C) = \frac{2 \times 1256}{51 \times 89} \][/tex]
3. Calculate the value of [tex]\(\sin(C)\)[/tex]:
[tex]\[\sin(C) \approx 0.5534258647279137\][/tex]
4. Determine the angle [tex]\(C\)[/tex] in radians:
[tex]\[ C = \arcsin(0.5534258647279137) \][/tex]
The inverse sine function (arcsin) will give us the angle in radians.
[tex]\[ C \approx 0.5864718281888074 \text{ radians} \][/tex]
5. Convert the angle from radians to degrees:
Use the conversion factor:
[tex]\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \][/tex]
So:
[tex]\[ C \approx 0.5864718281888074 \times \frac{180}{\pi} \text{ degrees} \][/tex]
[tex]\[ C \approx 33.6 \text{ degrees} \][/tex]
### Final Result:
The measure of the included angle [tex]\(C\)[/tex] in the triangle is approximately [tex]\(33.6\)[/tex] degrees, to the nearest tenth of a degree.
