The angles where sin(θ) = 0 in the interval [0, 2π] are θ = 0 and θ = π.

sin(θ) = 0 at θ = 0° and 180°.

Using the identity sin^2(θ) + cos^2(θ) = 1, we have cos^2(θ) = 1 - (1/√2)^2 = 1 - 1/2 = 1/2. Thus, cos(θ) = ±1/√2.

sin(θ) = 1/√2 at θ = 45°.

sin(θ) = 1/√2 at θ = 45° and θ = 225°.

sin(θ) = 1/√2 at θ = 45°.

Using the Pythagorean identity, cos(θ) = √(1 - sin²(θ)) = √(1 - (3/5)²) = 4/5.

Using the identity tan(θ) = sin(θ)/cos(θ) and cos(θ) = √(1 - sin^2(θ)), we find cos(θ) = 3/5. Thus, tan(θ) = (4/5)/(3/5) = 4/3.

The equation sin^(-1)(x) + cos^(-1)(x) = π/2 holds for all x in the domain of the functions, which is [-1, 1]. Therefore, x can be any value in this range.

Using the identity sin^(-1)(x) + cos^(-1)(x) = π/2, we can conclude that x can take any value in the range [-1, 1].

If sin^(-1)(x) = π/4, then x = sin(π/4) = √2/2.

Using the identity tan A = sin A / cos A, we can find sin A = tan A * cos A. Using the Pythagorean identity, we find sin A = 3/5.

Since tan θ = sin θ / cos θ and tan θ = 1, we have sin θ = cos θ. For θ = 45°, sin θ = 1/√2.

If tan(x) = 1, then sin(x) = cos(x). Therefore, sin(x) + cos(x) = 2sin(x) = 2(1/√2) = √2.

tan(45°) = 1, hence x = 45°.

Using the identity tan(x) = sin(x)/cos(x), we can find sin(x) = 3/5 after applying the Pythagorean theorem.

Using the identity tan(x) = sin(x)/cos(x), we can find sin(x) = 3/5 after calculating the hypotenuse using the Pythagorean theorem.

tan(θ) = 1 at θ = 45°.

Using the identity sin²(θ) + cos²(θ) = 1, we find sin(θ) = 3/5.

tan^(-1)(x) = π/4 implies that x = tan(π/4) = 1.

If the angle is 90 degrees, A · B = |A||B|cos(90) = 0.

A · B = |A||B|cos(60°) = √(2^2 + 3^2) * √(4^2 + 5^2) * 1/2 = √13 * √41 * 1/2 = 20.

Height = shadow * tan(angle) = 10 * √3 = 5√3 m.

Let the angles be 2x, 3x, and 4x. Then, 2x + 3x + 4x = 180 degrees. Thus, 9x = 180 degrees, x = 20 degrees. The largest angle is 4x = 80 degrees.

Let the angles be 2x, 3x, and 4x. Then, 2x + 3x + 4x = 180. So, 9x = 180, x = 20. The largest angle = 4x = 80 degrees.

Area = 1/2 * base * height => height = 30/(1/2 * 10) = 6.

Area = 1/2 * base * height. Therefore, 30 = 1/2 * 10 * height, height = 6 cm.

Area = 1/2 * base * height => 30 = 1/2 * 10 * height => height = 30 / 5 = 6.

Area = 1/2 * base * height. Thus, 30 = 1/2 * 10 * height. Height = 6.

Area = (1/2) * base * height. Therefore, 60 = (1/2) * 12 * height, height = 10 cm.