The solutions are x = π/4, 3π/4, 5π/4, and 7π/4.

The solutions are x = 0, π, and 2π.

sin(2x) = 0 implies 2x = nπ, thus x = nπ/2 for n ∈ Z.

The solutions are x = π/3 + 2nπ and x = 2π/3 + 2nπ.

Factoring gives sin(x)(sin(x) - 1) = 0, so sin(x) = 0 or sin(x) = 1. Thus, x = nπ or x = π/2 + 2nπ.

The area of an equilateral triangle is given by the formula (√3/4)a².

For a right triangle, the circumradius R = hypotenuse/2 = 13/2 = 6.5.

Circumradius R = (abc)/(4K), where K is the area. Area K = 24 (using Heron's formula). R = (6*8*10)/(4*24) = 5.

The circumradius R of a triangle can be calculated using the formula R = (abc)/(4 * Area). Here, Area = 84 cm², so R = (7 * 24 * 25)/(4 * 84) = 13.

The circumradius R of an equilateral triangle is given by R = a/(√3).

R = (abc) / (4 * Area). Area = 84, R = (7 * 24 * 25) / (4 * 84) = 12.

The circumradius R of a triangle is given by the formula R = abc/(4A), where A is the area of the triangle.

Using the formula r = A/s, where A is the area and s is the semi-perimeter. Area = 26 cm², s = 12 cm, so r = 26/12 = 4 cm.

Using the area formula, Area = 1/2 * base * height. First, find the area using Heron's formula, then use it to find the height.

The area of the triangle can be calculated as (1/2) * base * height. The area is 30 cm² (since it's a right triangle). Using area = (1/2) * 12 * height, we find height = 5 cm.

Using the area formula, Area = 1/2 * base * height. The area can also be calculated using Heron's formula, which gives 24. Thus, height = (2 * Area) / base = (2 * 24) / 10 = 4.8.

Median length m_a = 1/2 * √(2b^2 + 2c^2 - a^2) = 1/2 * √(2*8^2 + 2*10^2 - 6^2) = 1/2 * √(128 + 200 - 36) = 1/2 * √292 = 7.

By the double angle identity, 2sin(θ)cos(θ) = sin(2θ).

cos(0°) = 1.

cos(180°) = -1.

Using the double angle formula cos(2x) = 1 - 2sin^2 x. Since sin x = 1/2, we have cos(2x) = 1 - 2*(1/2)^2 = 1 - 2*(1/4) = 1 - 1/2 = 1/2.

Using the double angle formula cos(2θ) = 1 - 2sin^2 θ. Here, sin θ = 1/2, so cos(2θ) = 1 - 2(1/2)^2 = 1 - 2(1/4) = 1 - 1/2 = 1/2.

cos(90°) = 0.

cos(π/3) = 1/2.

sec(60°) = 1/cos(60°) = 1/(1/2) = 2.

sec(θ) = 1/cos(θ) = 1/(1/3) = 3.

sec(θ) = 1/cos(θ) = 1/(3/5) = 5/3.

The double angle formula states that sin 2θ = 2sin θ cos θ.

Using the double angle formula sin(2x) = 2sin x cos x. Since sin x = 1/2, cos x = √(1 - (1/2)^2) = √(3/4) = √3/2. Thus, sin(2x) = 2 * (1/2) * (√3/2) = √3/2.

Using the double angle formula sin(2θ) = 2sin θ cos θ. First, find cos θ using sin^2 θ + cos^2 θ = 1. cos θ = sqrt(1 - (1/3)^2) = sqrt(8/9) = 2sqrt(2)/3. Thus, sin(2θ) = 2 * (1/3) * (2sqrt(2)/3) = 4sqrt(2)/9.