To determine the length of the third side of the triangle with sides of lengths 2 and 5 and an angle of \( 60^\circ \) between them, we can use the Law of Cosines. The Law of Cosines is useful for finding the third side of a triangle when we know two sides and the included angle.
The formula for the Law of Cosines is given by:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\][/tex]
where \( a \) and \( b \) are the lengths of the given sides, \( C \) is the included angle, and \( c \) is the length of the third side opposite the angle \( C \).
Step-by-step solution:
1. Identify the given values:
- Side \( a = 2 \)
- Side \( b = 5 \)
- Angle \( C = 60^\circ \)
2. Convert the angle from degrees to radians: Since most trigonometric functions in mathematics are defined in terms of radians, we first convert \( 60^\circ \) to radians.
[tex]\[
\text{Angle in radians} = \frac{60 \times \pi}{180} = \frac{\pi}{3}
\][/tex]
Therefore:
[tex]\[
\cos(60^\circ) = \cos\left(\frac{\pi}{3}\right)
\][/tex]
3. Calculate \( \cos(60^\circ) \):
[tex]\[
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
\][/tex]
4. Apply the Law of Cosines:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\][/tex]
Substitute the values:
[tex]\[
c^2 = 2^2 + 5^2 - 2 \cdot 2 \cdot 5 \cdot \frac{1}{2}
\][/tex]
[tex]\[
c^2 = 4 + 25 - 10
\][/tex]
[tex]\[
c^2 = 19
\][/tex]
5. Find the length of side \( c \):
[tex]\[
c = \sqrt{19}
\][/tex]
Hence, the length of the third side of the triangle is:
[tex]\[
\boxed{\sqrt{19}}
\][/tex]
The correct answer is:
C. [tex]\( \sqrt{19} \)[/tex]
