# The sides of a right-angled triangle are 5cm, 12cm, 13cm. Find the tangent of the larger acute angle of this triangle.

It is known that in a right-angled triangle, the larger leg lies opposite the larger acute angle, and the smaller leg lies opposite the smaller acute angle.

According to the condition of the problem, the sides of this right-angled triangle are 5 cm, 12 cm and 13 cm.

Since the hypotenuse in a right-angled triangle is always larger than the legs of this triangle, then in this triangle the 13 cm side is its hypotenuse, and the 5 cm and 12 cm sides are its legs.

Consequently, against the larger acute angle of this right-angled triangle, the larger leg is 12 cm long.

Let us denote this acute angle by α. Applying the theorem of sines for the angle α, its opposite leg, right angle and hypotenuse opposite the right angle, we obtain the following relation:

12 / sin (α) = 13 / sin (90 °).

Since sin (90 °) = 1, we get;

sin (α) = 12/13.

Find the cosine of the larger acute angle of the given right triangle

Applying the cosine theorem for the angle β, we obtain the following relation:

5² + 13² – 2 * 5 * 13 * cos (α) = 12².

We find from this relation cos (α):

25 + 169 – 130 * cos (α) = 144;

194 – 130 * cos (α) = 144;

130 * cos (α) = 194 – 144;

130 * cos (α) = 50;

cos (α) = 50/130;

cos (α) = 5/13.

Find the tangent of the larger acute angle of this triangle

tg (α) = sin (α) / cos (α) = (12/13) / (5/13) = (12/13) * (13/5) = 12/5.

Answer: The tangent of the larger acute angle of this triangle is 12/5.