Explanation:
when you look at the graphic and the right-angled triangle :
sine is the length of the vertical leg (opposite of the associated angle) divided by the length of the Hypotenuse (the baseline opposite of the 90° angle), because the sine function values are defined for the norm circle with radius 1. for any larger circle (or triangle inscribed in such a larger circle), we get the basic trigonometric function values by measuring the sides and norming to a radius of 1 (dividing them by the length of the actual radius).
cosine is the horizontal leg (at the "bottom").
the same principle applies. the cosine value is the length of that leg divided by the length of the Hypotenuse.
we know from the given ratio for sine, that the length of the Hypotenuse is 6.
so, we know 2 sides of the right-angled triangle :
the vertical leg is 5, the Hypotenuse is 6.
now, we use Pythagoras to get the horizontal leg :
6² = 5² + leg²
36 = 25 + leg²
11 = leg²
leg = sqrt(11)
and so cosine of the angle is
sqrt(11)/6