To solve the problem of finding the length of the third side of the triangle, given sides \(a = 2\) and \(b = 3\) with an angle between them of \(60^\circ\), we can use the Law of Cosines. Here's a detailed, step-by-step solution:
1. Recall the Law of Cosines:
The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\) and included angle \(\theta\):
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(\theta)
\][/tex]
2. Substitute the given values:
- \(a = 2\)
- \(b = 3\)
- \(\theta = 60^\circ\)
3. Convert the angle to radians: Since cosine functions in trigonometry often require angles in radians:
[tex]\[
60^\circ = \frac{\pi}{3} \text{ radians}
\][/tex]
4. Calculate the cosine of 60 degrees:
[tex]\[
\cos\left(60^\circ\right) = 0.5
\][/tex]
5. Substitute the angle and side lengths into the Law of Cosines formula:
[tex]\[
c^2 = 2^2 + 3^2 - 2 \times 2 \times 3 \times 0.5
\][/tex]
6. Simplify the equation step-by-step:
[tex]\[
c^2 = 4 + 9 - 6
\][/tex]
[tex]\[
c^2 = 13 - 6
\][/tex]
[tex]\[
c^2 = 7
\][/tex]
7. Take the square root of both sides to solve for \(c\):
[tex]\[
c = \sqrt{7}
\][/tex]
Therefore, the length of the third side of the triangle is:
[tex]\[
\boxed{\sqrt{7}}
\][/tex]
