The leg opposite to the 60-degree angle of this right-angled triangle is 3.

The leg opposite to the 60-degree angle of this right-angled triangle is 3. find the hypotenuse, the second leg and the acute angle of this triangle

Let us denote by c the length of the hypotenuses of this right-angled triangle.

In the initial data for this task, it is reported that one of the acute angles of this triangle is 60 °, and the leg opposite to the angle is 3, therefore, applying the theorem of sines, we can find the length of the hypotenuse of this triangle:

c = sin (90 °) * 3 / sin (60 °) = 3 / (√3 / 2) = 3 * 2 / √3 = 6 / √3 = 6√3 / 3 = 2√3.

Using the Pythagorean theorem, we find the second leg:

√ ((2√3) ^ 2 – 3 ^ 2) = √ (12 – 9) = √3.

Since the sum of all the angles of any triangle is 180 °, we find the second acute angle of this triangle:

180 – 90 – 60 = 90 – 60 = 30 °.

Answer: the hypotenuse is 2√3, the second leg is √3, the second acute angle is 30 °.