To determine the length of the third side of a triangle when two sides and the included angle are known, 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]\( c \)[/tex] is the length of the third side we need to find, [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the given sides, and [tex]\( C \)[/tex] is the included angle.
Given:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( C = 60^\circ \)[/tex]
First, we convert the angle from degrees to radians since the cosine function generally uses radians in mathematical calculations. However, we will assume we can directly use the degrees for conceptual understanding here:
Using [tex]\( C = 60^\circ \)[/tex] and knowing that [tex]\( \cos(60^\circ) = \frac{1}{2} \)[/tex]:
Let's plug in the values into the Law of Cosines formula:
[tex]\[ c^2 = 3^2 + 4^2 - 2 \cdot 3 \cdot 4 \cdot \cos(60^\circ) \][/tex]
[tex]\[ c^2 = 9 + 16 - 2 \cdot 3 \cdot 4 \cdot \frac{1}{2} \][/tex]
[tex]\[ c^2 = 9 + 16 - 12 \][/tex]
[tex]\[ c^2 = 25 - 12 \][/tex]
[tex]\[ c^2 = 13 \][/tex]
Next, we find the length of [tex]\( c \)[/tex] by taking the square root of both sides of the equation:
[tex]\[ c = \sqrt{13} \][/tex]
So, the length of the third side of the triangle is [tex]\( \sqrt{13} \)[/tex].
Therefore, the correct answer is:
C. [tex]\( \sqrt{13} \)[/tex]
