The legs of a right-angled triangle are equal to 3√91 and 9. Find the sine of the smallest angle of this triangle.

The sine of an acute angle of a right-angled triangle is the ratio of the leg opposite to a given angle to the hypotenuse.

1) Let us find the hypotenuse by the Pythagorean theorem. The square of the hypotenuse is equal to the sum of the squares of the legs.

a = 9; b = 3√91; c -?

c ^ 2 = a ^ 2 + b ^ 2;

c ^ 2 = 9 ^ 2 + (3√91) ^ 2;

c ^ 2 = 81 + 9 * 91;

c ^ 2 = 81 + 819;

c ^ 2 = 900;

c = 30.

2) Find the sine of the smallest angle of the triangle. The smallest angle lies opposite the smaller side. The smallest side will have a side of length 9.

sin a = a / c;

sin a = 9/30 = 3/10 = 0.3.

Answer. 0.3.