The Spanish for syrup, jarabe, comes from the same root as the English: the Persian/Arabic sharab, which means “a drink, or wine”.
The drastically different (at least superficially) words are explained by the sh- and related (such as, sy- ) sounds changing to the Arabic-sounding j- sound in Spanish — but not English.
Thus, the j-r-b of jarabe maps to the sy-r-p of syrup.
Etapa (Spanish for “stage, level”) comes from old Dutch word (remember, the whole Spanish-Netherlands 80 years war? They did influence each other a lot!) stapel meaning, “deposit; store.”
The English staple comes from the Old German stapulaz (“pillar”) — from which we also get the Dutch stapel and then the Spanish etapa!
But how did a word meaning “pillar” become “stage” or “staple”? Well, a pillar holds up the next level — the next stage! (Think of floors in a building as being stages of development. Ultimately we reach the penthouse!). Or think about the pillar — that which holds everything else up so it doesn’t fall — is the staple of the building, the most basic building block, to ensure it doesn’t collapse!
We can see the t-p root in both the English and Spanish words.
The Spanish mancha (“spot” or “stain”) comes from the Latin for the same, macula.
From the Latin macula, we get the English… immaculate — which literally means (knowing the negation prefix of im-) “without a stain.” So the immaculate conception truly was perfect!
How this sound changed was interesting: often Latin words with a ct- or cl- or other hard letters after a c- sound turn into a suave ch in Spanish. For a distant example, see duct and ducha, or nocturnal and noche. (The ct- is much more common than the cl-, but they’re cousins!) Thus, we can see the m-ch of mancha mapping to the (im-)m-cl of immaculate.
The Spanish lágrima (“tear”) comes from the Latin Lacrima, meaning the same.
From the same root we get the English… lacrimal sac. In case you forgot our high school biology class, that’s the bit by your eye that creates… tears.
The l-c-r of lacrimal sac maps to the l-g-r of lágrima.
The usual Spanish word for “name”, nombre, is very closely related to the English word nominal, in an interesting way. Not only does nominally mean “relating to the name”, but there is an interesting etymological pattern between the words.
Latin words with an m-n sound usually turned the m-n into an mbr sound as Latin evolved into Spanish. Thus, we see curious patterns like hominem becoming hombre, and famine and hambre being closely linked.
The same pattern applies here. The Latin nominalis turned into the Spanish nombre and the English nominal — thus the n-m-n of nominal maps exactly to the n-mbr of nombre!