Step 1: Understanding the Concept:
When two sides and the included angle of a triangle are provided, we use the Law of Cosines to determine the length of the third side.
The Law of Cosines is a generalization of the Pythagorean theorem that applies to all types of triangles.
In this specific case, the sides \( a \) and \( b \) are known, and the angle \( C \) between them is given in radians.
Step 2: Key Formula or Approach:
The Law of Cosines for side \( c \) is:
\[ c^2 = a^2 + b^2 - 2ab \cos C \]
Step 3: Detailed Explanation:
From the problem:
\( a = 1 \)
\( b = \sqrt{3} \)
\( C = \frac{\pi}{6} = 30^\circ \)
We know that the cosine of \( 30^\circ \) is \( \frac{\sqrt{3}}{2} \).
Substitute these values into the formula:
\[ c^2 = (1)^2 + (\sqrt{3})^2 - 2(1)(\sqrt{3}) \cos(30^\circ) \]
\[ c^2 = 1 + 3 - 2(1)(\sqrt{3}) \left( \frac{\sqrt{3}}{2} \right) \]
Simplify the expression:
\[ c^2 = 4 - 2 \left( \frac{\sqrt{3} \times \sqrt{3}}{2} \right) \]
The factor of 2 in the numerator and denominator cancels out:
\[ c^2 = 4 - (\sqrt{3} \times \sqrt{3}) \]
\[ c^2 = 4 - 3 \]
\[ c^2 = 1 \]
Taking the square root to find the side length (which must be positive):
\[ c = \sqrt{1} = 1 \]
The triangle has sides 1, 1, and \( \sqrt{3} \). This is an isosceles triangle with angles \( 30^\circ, 30^\circ, 120^\circ \).
Step 4: Final Answer:
The length of the third side \( c \) is 1.
This is Option (C).