Step 1: Understanding the Concept:
This problem requires evaluating a limit involving powers of trigonometric functions and then using coordinate geometry to find the area of a triangle. The limit of \( r^n \) as \( n \to \infty \) depends on the magnitude of \( r \).
Step 2: Key Formula or Approach:
1. Evaluate \( a \) and \( b \) using the properties of \( \lim_{n \to \infty} r^n \).
2. Use the vertices formula for the area of a triangle: \( \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \).
Step 3: Detailed Explanation:
- Evaluating \( a \):
Given \( x = n\pi \). Then \( \cos x = \cos(n\pi) = (-1)^n \).
The term is \( \cos^{2n} x = ((-1)^n)^{2n} = (-1)^{2n^2} \). Since \( 2n^2 \) is always an even number, \( (-1)^{2n^2} = 1 \).
So, \( a = \lim_{n \to \infty} 1 = 1 \).
- Evaluating \( b \):
Given \( x \neq m\pi \). For these values, \( |\cos x|<1 \).
Let \( r = \cos^2 x \). Since \( |\cos x|<1 \), we have \( 0 \le r<1 \).
The limit is \( b = \lim_{n \to \infty} r^n \). Since \( |r|<1 \), the limit is 0.
So, \( b = 0 \).
The vertices of the triangle are:
\( A(1, 0) \), \( B(-2, 1) \), and \( C(2, 1) \).
Notice that vertices \( B \) and \( C \) lie on the same horizontal line \( y = 1 \).
The base length of the triangle is the distance between \( B \) and \( C \):
\[ \text{Base} = |2 - (-2)| = 4 \]
The height of the triangle is the vertical distance from vertex \( A(1, 0) \) to the line \( y = 1 \):
\[ \text{Height} = |1 - 0| = 1 \]
Area \( = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 4 \times 1 = 2 \).
Step 4: Final Answer:
The area of the triangle is 2.