C^2 = C * C = [[1*1 + 2*3, 1*2 + 2*5], [3*1 + 5*3, 3*2 + 5*5]] = [[11, 16], [18, 35]].

The total number of subsets is 2^4 = 16. Proper subsets exclude the set itself, so there are 16 - 1 = 15 proper subsets.

The proper subsets of a set exclude the set itself. For C = {1, 2}, the proper subsets are {∅}, {1}, and {2}, totaling 3 subsets.

The set C = {2, 3, 5, 7}. The power set of C includes all subsets, including the empty set, so it is {{}, {2}, {3}, {5}, {7}, {2, 3}, {2, 5}, {2, 7}, {3, 5}, {3, 7}, {5, 7}, {2, 3, 5}, {2, 3, 7}, {2, 5, 7}, {3, 5, 7}, {2, 3, 5, 7}}.

The even integers less than 10 are 0, 2, 4, 6, and 8.

The power set of an infinite set is also infinite. Since C contains all even integers, its power set is infinite.

The set {2, 4, 6} contains only even integers, making it a subset of C.

C's investment = 20000, D's investment = 30000. Total investment = 50000. D's share = (30000/50000) * 50000 = $30,000.

C's share = (20000 / (20000 + 30000)) * 50000 = (20000 / 50000) * 50000 = $20,000.

The angle A for which cos A = 1/2 is 60°.

Using the identity tan A = sin A / cos A, we find sin A = √(1 - cos²A) = √(1 - (5/13)²) = 12/13. Thus, tan A = (12/13) / (5/13) = 12/5.

Using the Pythagorean identity, sin B = √(1 - cos²B) = √(1 - 0.6²) = √(1 - 0.36) = √(0.64) = 0.8.

Cosine of 60° is 1/2, hence angle B = 60°.

Using the identity sin^2 B + cos^2 B = 1, we have sin B = sqrt(1 - (1/2)^2) = sqrt(3/4) = √3/2.

Using the Pythagorean identity, sin B = √(1 - cos²B) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13.

The angle C for which cos C = 0.5 is 60°.

cos(x) = 0 at x = 90° + k*180°, where k is any integer.

tan(x) = sin(x)/cos(x) is undefined when cos(x) = 0.

cos(x) = 0 at x = π/2 + nπ, where n is any integer.

cos(30°) = 0.5.

Using the identity sin^2(x) + cos^2(x) = 1, we have sin^2(x) = 1 - (1/2)^2 = 1 - 1/4 = 3/4. Therefore, sin(x) = √(3/4) = √3/2.

The angles where cos(α) = 1/2 in the interval [0, 2π] are α = π/3 and α = 5π/3.

cos(90°) = 0, so θ = 90°

cos(60°) = 0.5, so θ = 60°

cos(60°) = 0.5, so θ = 60°.

cos(30°) = 0.5, so θ = 30°.

sin²(θ) + cos²(θ) = 1; sin²(θ) = 1 - 0.6² = 0.64; sin(θ) = √0.64 = 0.8.

cos(45°) = √2/2 ≈ 0.707, so θ = 45°.

cos(θ) = 1/2 at θ = 30° and 150°.

cos(θ) = 1/2 at θ = 60° and θ = 300°.