According to Dictionary.com
Minimal is an adjective and Minimum is a noun. They have different meanings and purposes.
constituting a minimum:
a minimal mode of transportation.
barely adequate or the least possible:
noun, plural minimums, minima [min-uh-muh] (Show IPA)
the least quantity or amount possible, assignable, allowable, or the like.
the lowest speed permitted on a highway.
the lowest amount, value, or degree attained or recorded.
an arbitrary amount set by a restaurant, nightclub, etc., as the least amount to be charged each person for food and drink.
Compare cover charge.
Also called relative minimum, local minimum. the value of a function at a certain point in its domain, which is less than or equal to the values at all other points in the immediate vicinity of the point.
Compare absolute minimum.
the point in the domain at which a minimum occurs.
The word Minimal comes from the word minimum consisting of synonyms nominal and minimum.
The word Minimum originates from the Latin word minimus meaning smallest or least. Minimum's synonyms consist of minimal and merest.
Based on your sentences above, The correct way of using the words minimum and minimal would be as follows:
The minimal angle-correct because it's being used as an adjective
The minimum angle-incorrect usage using a noun as an adjective
(The same would follow for optimum/optimal)