Certainly! To find the length of the third side of a triangle given two sides and the included angle, we can use the Law of Cosines. The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two given sides, [tex]\( C \)[/tex] is the included angle, and [tex]\( c \)[/tex] is the length of the third side we want to find.
Given:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( C = 60^\circ \)[/tex]
First, we need to convert the angle from degrees to radians since trigonometric functions typically use radians.
[tex]\[ 60^\circ = \frac{\pi}{3} \text{ radians} \][/tex]
This conversion results in approximately [tex]\( 1.0472 \)[/tex] radians.
Next, we plug the values into the Law of Cosines formula:
[tex]\[ c^2 = 2^2 + 3^2 - 2 \cdot 2 \cdot 3 \cdot \cos\left(\frac{\pi}{3}\right) \][/tex]
Recall that [tex]\( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)[/tex]:
[tex]\[ c^2 = 4 + 9 - 2 \cdot 2 \cdot 3 \cdot \frac{1}{2} \][/tex]
[tex]\[ c^2 = 4 + 9 - 6 \][/tex]
[tex]\[ c^2 = 7 \][/tex]
To find [tex]\( c \)[/tex], take the square root of both sides:
[tex]\[ c = \sqrt{7} \][/tex]
Therefore, the length of the third side of the triangle is [tex]\( \sqrt{7} \)[/tex].
So, the correct answer is:
A. [tex]\( \sqrt{7} \)[/tex]
