In a right triangle, one angle measures x°, where  sin (x°) = 12/13. What is sin (90° - x°), expressed as a fraction in simplest form?

In a right triangle, the sum of the angles is always 90°. Therefore, if one angle is x°, the other angle is 90° - x°.

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

Given that sin(x°) = 12/13, this means that the length of the opposite side to angle x° is 12 and the length of the hypotenuse is 13.

In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Since the cosine of an angle is equal to the sine of its complement, we can say that cos(x°) = sin(90° - x°).

Therefore, sin(90° - x°) = cos(x°).

We know that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

So, if the length of the opposite side is 12 and the length of the hypotenuse is 13, the length of the adjacent side can be found using the Pythagorean theorem:

(Adjacent side)^2 = (Hypotenuse)^2 - (Opposite side)^2 (Adjacent side)^2 = 13^2 - 12^2 (Adjacent side)^2 = 169 - 144 (Adjacent side)^2 = 25 Adjacent side = sqrt(25) Adjacent side = 5

Therefore, cos(x°) = Adjacent side / Hypotenuse = 5 / 13.

So, sin(90° - x°) = cos(x°) = 5/13.

In a right triangle, the sum of the angles is always 90°. If one angle is x°, then the other angle is 90° - x°.

The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Given that sin(x°) = 12/13, this means that the side opposite angle x° is 12 units long and the hypotenuse is 13 units long.

In a right triangle, the cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Since the side adjacent to angle x° is also the side opposite to angle 90° - x°, cos(x°) is also equal to sin(90° - x°).

We know that in a right triangle, sin²(x) + cos²(x) = 1 (this is known as the Pythagorean trigonometric identity).

So, we can solve for cos(x°) = sqrt[1 - sin²(x)] = sqrt[1 - (12/13)²] = sqrt[1 - 144/169] = sqrt[25/169] = 5/13.

Therefore, sin(90° - x°) = cos(x°) = 5/13.