To find the length of the third side of a triangle when we know the lengths of two sides and the angle between them, we can use the Law of Cosines. The Law of Cosines formula is given by:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \][/tex]
where:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two known sides,
- [tex]\( \theta \)[/tex] is the included angle,
- [tex]\( c \)[/tex] is the length of the third side.
In this problem, we know:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( \theta = 60^\circ \)[/tex]
First, we need to convert the angle from degrees to radians, since the cosine function in the Law of Cosines formula often requires the angle in radians.
[tex]\[ \theta = 60^\circ = \frac{\pi}{3} \][/tex]
Next, we apply 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]
We know that:
[tex]\[ \cos\left(\frac{\pi}{3}\right) = 0.5 \][/tex]
Therefore, substituting these values into the formula gives:
[tex]\[ c^2 = 4 + 9 - 2 \cdot 2 \cdot 3 \cdot 0.5 \][/tex]
[tex]\[ c^2 = 4 + 9 - 6 \][/tex]
[tex]\[ c^2 = 7 \][/tex]
To find [tex]\( c \)[/tex], we take the square root of both sides:
[tex]\[ c = \sqrt{7} \][/tex]
Thus, the length of the third side of the triangle is:
[tex]\[ \boxed{\sqrt{7}} \][/tex]
