To find the length of the third side of a triangle when you know the lengths of two sides and the angle between them, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], where [tex]\(c\)[/tex] is the side opposite the angle [tex]\(\gamma\)[/tex]:
[tex]\[ c^2 = a^2 + b^2 - 2ab\cos(\gamma) \][/tex]
Given the problem:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 5\)[/tex]
- [tex]\(\gamma = 60^\circ\)[/tex]
First, we need to convert the angle from degrees to radians because the cosine function in the steps generally works with angles in radians.
[tex]\[\gamma = 60^\circ = \frac{\pi}{3} \text{ radians} (approximately 1.0472 \text{ radians})\][/tex]
Now we apply the Law of Cosines:
[tex]\[ c^2 = 2^2 + 5^2 - 2 \cdot 2 \cdot 5 \cdot \cos(60^\circ) \][/tex]
We know that [tex]\(\cos(60^\circ) = \frac{1}{2}\)[/tex]. Substituting this into the equation:
[tex]\[ c^2 = 4 + 25 - 2 \cdot 2 \cdot 5 \cdot \frac{1}{2} \][/tex]
[tex]\[ c^2 = 4 + 25 - 10 \][/tex]
[tex]\[ c^2 = 19 \][/tex]
To find the length of the third side [tex]\(c\)[/tex], we take the square root of both sides:
[tex]\[ c = \sqrt{19} \][/tex]
Hence, the length of the third side of the triangle is [tex]\(\sqrt{19}\)[/tex].
Therefore, the correct answer is:
D. [tex]\(\sqrt{19}\)[/tex]
