Origins
The prefix 'trans-' is a Latin-derived element meaning 'across,' 'over,' 'beyond,' or 'through.' It descends from Latin 'trans' (across), originally a participial form from Proto-Indo-European *terh₂- meaning 'to cross over, pass through.' The same PIE root produced Sanskrit 'tirás' (across, through), Old Irish 'tar' (across), and — through Germanic — English 'through' and its cognates German 'durch' and Gothic 'þairh.' Latin 'trans' and English 'through' are etymological siblings separated by thousands of years of divergent development.
In Latin, 'trans-' was productive on verbs of motion and transformation: 'transire' (to go across — whence 'transit,' 'transient'), 'transferre' (to carry across — whence 'transfer'), 'transmittere' (to send across — whence 'transmit'), 'transformare' (to change shape — whence 'transform'), 'transportare' (to carry over — whence 'transport'). The prefix consistently denoted motion from one side to the other, change of state, or transition through.
'Trans-' entered English in the fourteenth and fifteenth centuries in learned borrowings from Latin and Anglo-Norman French: 'transact,' 'transcend,' 'transfer,' 'transfigure,' 'transfix,' 'transform,' 'transfuse,' 'transgress,' 'transient,' 'transit,' 'translate,' 'translucent,' 'transmigrate,' 'transmit,' 'transmute,' 'transparent,' 'transpire,' 'transplant,' 'transport,' 'transpose,' 'transverse.' In many of these the prefix is semi-transparent to modern speakers: 'transport' is recognisably 'carry across,' 'translate' is 'carry over' (from 'trans-' + 'latus,' carried), 'transmit' is 'send across.' In others the connection is lost: 'trespass' (Old French 'trespasser,' to pass across / transgress) and 'trance' (from Old French 'transe,' a passing — extreme state — originally from Latin 'transire,' to go across) retain the prefix in disguised form.
Word Formation
From the late nineteenth and especially the twentieth centuries, 'trans-' became one of the most productive geographic and technical prefixes in English. Geographic: 'transatlantic' (1779), 'transcontinental' (1853), 'transpacific' (1898), 'transnational' (1921), 'transoceanic,' 'transpolar,' 'trans-European,' 'trans-Siberian.' Technical and scientific: 'transistor' (1948, portmanteau of 'transfer resistor'), 'transfusion,' 'transduce,' 'transcribe,' 'transcript,' 'transmission,' 'transducer,' 'transceiver,' 'transponder,' 'transgene,' 'transcriptome.' In chemistry, 'trans-' is one of two geometric isomer prefixes (paired with 'cis-'): 'trans-fatty acid,' 'trans-isomer.'
In the late twentieth century a new productive use of 'trans-' emerged in social and political vocabulary: 'transgender' (coined 1965, popularised 1990s), 'transsexual' (earlier, 1950s), 'transvestite' (1910, from German), 'trans-man,' 'trans-woman.' By the 2010s, 'trans' had become a widely used freestanding adjective and noun referring to transgender people ('trans rights,' 'trans community,' 'she is trans'). This is a rare example of a bound Latin prefix becoming a freestanding English word in the space of a few decades, paralleling the similar development of 'ex-' in the nineteenth century.
Hyphenation of 'trans-' is variable and register-dependent. Naturalised words never take a hyphen: 'transport,' 'translate,' 'transform.' Transparent coinages usually do when followed by a proper noun or to avoid ambiguity: 'trans-Atlantic,' 'trans-European,' 'trans-Pacific.' In recent usage for gender identity, 'trans-' with a hyphen is less common than the solid form or standalone 'trans.'
Spelling and Pronunciation
Pronunciation varies. The main form is /træns/ (American) or /trɑːns/ (British) when the prefix is stressed or the base starts with a voiceless consonant: 'transfer,' 'transport,' 'transcript.' Before voiced consonants or in fast speech it is often /trænz/: 'transit,' 'transition,' 'translate.' In British English the vowel can be the broad /ɑː/ rather than /æ/; American English almost always uses /æ/.
In chemistry and biology, 'trans-' has precise technical meanings. In organic chemistry, 'trans-' indicates a geometric isomer where two substituents lie on opposite sides of a double bond or ring (paired with 'cis-,' meaning same side). 'trans-2-butene,' 'trans-fatty acid.' In genetics, 'trans-' describes regulatory action across loci ('trans-acting factor' versus 'cis-acting element'). These technical uses are precise and not interchangeable with other 'cross' prefixes.
The difference between 'trans-' and the related prefix 'cross-' is useful to note. Both can indicate 'across,' but 'cross-' (native Germanic, from Old Norse 'kross') tends to denote perpendicular or transverse motion within a bounded space ('cross-section,' 'cross-country,' 'cross-reference'), while 'trans-' tends to denote traversal from one bounded region to another ('transcontinental,' 'transatlantic,' 'transnational'). In many cases both prefixes are possible with different shades of meaning.
Legacy
Representative 'trans-' words include: transact, transaction, transatlantic, transcend, transcendent, transcontinental, transcribe, transcript, transduce, transfer, transfiguration, transfix, transform, transformation, transfuse, transfusion, transgress, transient, transistor, transit, transition, transitive, translate, translation, translucent, transmigrate, transmission, transmit, transmute, transparent, transpire, transplant, transport, transportation, transpose, transverse, transvestite, transgender, trans-fat. Productive modern: trans-European, trans-Pacific, transnational, trans-Siberian, trans-polar, trans-species, transhumanism.