To find the area of triangle ATUV given two sides and the included angle, we can use the formula for the area of a triangle when you know two sides and the included angle between them. This formula is:
[tex]\[ \text{Area} = \frac{1}{2} uv \sin(M) \][/tex]
where:
- [tex]\( u \)[/tex] is one side of the triangle,
- [tex]\( v \)[/tex] is the other side,
- [tex]\( M \)[/tex] is the angle between sides [tex]\( u \)[/tex] and [tex]\( v \)[/tex].
Here are the given values:
- [tex]\( M = 68^\circ \)[/tex]
- [tex]\( u = 5 \)[/tex] cm
- [tex]\( v = 6 \)[/tex] cm
Step-by-step solution:
1. Convert the angle [tex]\( M \)[/tex] from degrees to radians. Since trigonometric functions in mathematical formulas generally use radians, we must convert degrees to radians by using the conversion factor:
[tex]\[ 1^\circ = \frac{\pi}{180} \][/tex]
So:
[tex]\[ M \text{ (in radians)} = 68^\circ \times \frac{\pi}{180} \approx 1.1868 \text{ radians} \][/tex]
2. Use the sine of the angle [tex]\( M \)[/tex]. We need to find [tex]\( \sin(68^\circ) \)[/tex]. If you use a calculator, it will give you approximately:
[tex]\[ \sin(68^\circ) \approx 0.9272 \][/tex]
3. Plug the values into the area formula:
[tex]\[ \text{Area} = \frac{1}{2} \times 5 \text{ cm} \times 6 \text{ cm} \times \sin(68^\circ) \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times 5 \times 6 \times 0.9272 \][/tex]
4. Compute the product:
[tex]\[ \text{Area} = \frac{1}{2} \times 5 \times 6 \times 0.9272 \approx \frac{1}{2} \times 5 \times 6 \times 0.9272 = 13.908 \][/tex]
5. Round the result to the nearest tenth:
[tex]\[ \text{Area} \approx 13.9 \, \text{square centimeters} \][/tex]
Thus, the area of triangle ATUV is approximately [tex]\( 13.9 \)[/tex] square centimeters.
