Find the perimeter of a right-angled triangle if you are given leg a and the opposite corner a.

Let’s denote by b the length of the second leg, and by c – the length of the hypotenuse of this right-angled triangle.

In the initial data for this task, it is reported that the value of the angle opposite the leg a is equal to α.

Since the sum of the angles of any triangle is 180 °, the value of the angle opposite the leg b is π – π / 2 – α = π / 2 – α.

Applying the sine theorem, we find b and c:

b = a * sin (π / 2 – α) / sin (α) = a * cos (α) / sin (α) = a * ctg (α);

c = a * sin (90 °) / sin (α) = a * 1 / sin (α) = a / sin (α).

Knowing the lengths of all sides, we find the perimeter of the triangle:

a + b + c = a + a * ctg (α) + a / sin (α) = a * (1 + ctg (α) + 1 / sin (α)).

Answer: a * (1 + ctg (α) + 1 / sin (α)).