All 3 Uses of

divine
in
Henry IV, Part 2

Yet speak, Morton; Tell thou an earl his divination lies, And I will take it as a sweet disgrace And make thee rich for doing me such wrong.†
Scene 1.1 *

divination = the act of predicting or discover something supernaturally (as if by magic)

Of forged rebellion with a seal divine
Scene 4.1 *

divine = royal (or emanating from God)

You, lord archbishop, Whose see is by a civil peace maintain'd, Whose beard the silver hand of peace hath touch'd, Whose learning and good letters peace hath tutor'd, Whose white investments figure innocence, The dove and very blessed spirit of peace, Wherefore you do so ill translate yourself Out of the speech of peace that bears such grace, Into the harsh and boisterous tongue of war; Turning your books to graves, your ink to blood, Your pens to lances and your tongue divine To a loud trumpet and a point of war?†
Scene 4.1

Definitions:

(1) (divine as in: to forgive is divine)
wonderful; or god-like or coming from God
(2) (divine as in: divined from tea leaves)
to discover or predict something supernaturally (as if by magic)
(3) (divine as in: divined through intuition)
to discover or guess something -- usually through intuition or reflection
(4) (meaning too rare to warrant focus)
meaning too rare to warrant focus:

In the time of Shakespeare, divine was sometimes used as a noun to reference a priest or a person of the church. (To remember that sense, think of the clergyman as having come from God).

Divinity typically refers to a god or to a school of religion, but on rare occasions, it refers to the name of a kind of soft white candy. To remember that sense, you might think of it as tasting divine/wonderful.