In a 30-60-90 triangle, the side opposite the 30° angle (let's call it "a") is half the length of the hypotenuse (let's call it "c"), and the side opposite the 60° angle (let's call it "b") is √3 times the length of the side opposite the 30° angle:

a = c/2
b = a√3

Now we know that the hypotenuse has a length of 16 cm:

a = 16/2
a = 8 cm

Now we can find the length of side b:

b = 8√3 cm

Since the triangle is a right triangle with one angle of 60°, the other acute angle must be 30°. The triangle is a 30-60-90 triangle. In a 30-60-90 triangle, the side adjacent to the 60° angle (let's call it "a") is half the length of the hypotenuse (let's call it "c").

In this case, the hypotenuse has a length of 17 cm. Therefore, the length of side a (adjacent to angle 60°) is 17/2 = 8.5 cm.

In a right triangle, the opposite side length (b) divided by the adjacent side length (a) gives the tangent of the angle (θ). In our case, we have:

tan(30°) = b / (6 cm)

We know that the tangent of 30° is equal to 1/√3. So, we have:

1/√3 = b / (6 cm)

To find the length of side b, we can multiply both sides by 6 cm:

b = 2√3 cm

Since the triangle is a right triangle with one angle of 30°, the other acute angle must be 60°. The triangle is a 30-60-90 triangle. In a 30-60-90 triangle, the side opposite the 30° angle (let's call it "a") is half the length of the hypotenuse (let's call it "c"):

a = c/2

Now we know that the hypotenuse has a length of 8 cm:

a = 8/2
a = 4 cm

First, find the value of cos(a) using the Pythagorean identity:cos²(a) + sin²(a) = 1
cos²(a) = 1 - sin²(a)
cos²(a) = 1 - (3/5)²
cos²(a) = 1 - 9/25
cos²(a) = 16/25
cos(a) = ±4/5Since a is an angle in a right triangle, the cosine must be positive. Therefore, cos(a) = 4/5.Now, find the length of the hypotenuse using the definition of sine:sin(a) = opposite / hypotenuse
3/5 = 12 / hypotenuseSolve for the hypotenuse:hypotenuse = 12 / (3/5) = 20 unitsSo, the value of cos(a) is 4/5 and the length of the hypotenuse is 20 units.