The angle at the apex of the axial section of the cone is alpha, and the distance from the center

The angle at the apex of the axial section of the cone is alpha, and the distance from the center of the base to the generatrix of the cone is a. Find the area of the lateral surface of the cone.

The axial section of the cone is an isosceles triangle, the base of which is the diameter of the base of the cone, and the lateral sides of the cone.
The lateral surface area of ​​the cone:
S = π * R * l,
where R is the radius of the base, l is the generator.
1. △ ASB – axial section of the cone: SA = SB = l, ∠ASB = α, SO – height △ -ka (as well as bisector and median) ⇒ AO = BO = AB / 2 = R, ∠ASO = ∠BSO = ∠ASB / 2 = α / 2.
From △ ASO by Sine Theorem:
AO / sin∠ASO = SA / sin∠SOA;
R / sin (α / 2) = l / sin90 °;
R / sin (α / 2) = l / 1;
R / sin (α / 2) = l;
R = l * sin (α / 2).
2. Consider △ BSO.
According to the condition OH = a – the height drawn to the generatrix (that is, the height drawn and the vertices of the right angle to the hypotenuse).
From the height properties of a right triangle:
OH = (SO * BO) / SB.
SO = (OH * SB) / BO (proportional).
SO = (a * l) / (l * sin (α / 2));
SO = a / sin (α / 2).
By the Pythagorean theorem:
SB = √ (SO² + BO²);
l = √ ((a / sin (α / 2)) ² + (l * sin (α / 2)) ²) = √ (a² / sin² (α / 2) + l² * sin² (α / 2)) = √ ((a² + l² * sin² (α / 2) * sin² (α / 2)) / sin² (α / 2)) = √ ((a² + l² * sin⁴ (α / 2)) / sin² (α / 2 )).
l² = ((a² + l² * sin ⁴ (α / 2)) / sin² (α / 2));
l² * sin² (α / 2) = a² + l² * sin⁴ (α / 2) (by proportion);
l² * sin² (α / 2) – l² * sin⁴ (α / 2) = a²;
l² * (sin² (α / 2) – sin⁴ (α / 2)) = a²;
l² = a² / (sin² (α / 2) – sin⁴ (α / 2) (by proportion);
l² = a² / (sin² (α / 2) * (1 – sin² (α / 2));
l² = a² / (sin² (α / 2) * cos² (α / 2) (by the basic trigonometric identity);
l = √ (a² / (sin² (α / 2) * cos² (α / 2));
l = a / (sin (α / 2) * cos (α / 2));
l = (2 * a) / (2 * sin (α / 2) * cos (α / 2)) (double angle sine formula);
l = (2 * a) / sinα.
3. Thus, the area of ​​the lateral surface of the cone is:
S = π * R * l = π * (2 * a) / sinα * sin (α / 2) * (2 * a) / sinα = (π * 4 * a² * sin (α / 2)) / sinα².
Answer: S = (π * 4 * a² * sin (α / 2)) / sinα².