
eingespannt
busy
Colloquial for very busy or occupied by many tasks and obligations.
Ich kann heute Abend nicht kommen, ich bin beruflich total eingespannt.
I can't come this evening; I'm totally tied up with work.
busy / clamped / harnessed
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busy
Colloquial for very busy or occupied by many tasks and obligations.
Ich kann heute Abend nicht kommen, ich bin beruflich total eingespannt.
I can't come this evening; I'm totally tied up with work.

clamped
Technically or mechanically fixed or clamped in a device.
Das Werkstück muss fest in den Schraubstock eingespannt werden, bevor man es bearbeitet.
The workpiece must be clamped firmly in the vise before you work on it.

harnessed
Refers to an animal, usually a horse or an ox, that has been put into a harness in order to pull a carriage, a plow, or another device.
Vor die Kutsche waren zwei prächtige Pferde eingespannt.
Two magnificent horses were harnessed to the carriage.
'Eingespannt' is the past participle of the separable verb 'einspannen' (to clamp in; to harness). The basic meaning is to put something firmly into a device. This idea connects all meanings: you are 'eingespannt' (tied up) with work, a workpiece is 'eingespannt' (clamped) in a vise, and a horse is 'eingespannt' (harnessed) to a carriage.
Der Techniker spannt die Platte in die Maschine ein.
The technician clamps the plate into the machine.
The context reveals the meaning. If people are being discussed, it means 'busy'. If a tool or material is being discussed, it means 'clamped'. If an animal is being discussed, it means 'harnessed'.
Mein Kollege ist momentan sehr eingespannt und hat keine Zeit.
My colleague is very busy at the moment and has no time.
In the meaning 'very busy', 'eingespannt' is mainly used colloquially. It sounds natural in everyday conversations.
Sorry, ich kann am Wochenende nicht helfen, da bin ich privat voll eingespannt.
Sorry, I can't help on the weekend; I'm completely tied up with personal commitments then.
When it means 'busy', 'eingespannt' is often combined with adverbs like 'beruflich' (professionally / for work), 'zeitlich' (time-wise), 'stark' (heavily), or 'total' (totally) to describe the reason or intensity.
Sie ist beruflich stark eingespannt und reist sehr viel.
She is heavily tied up with work and travels a lot.
active / employed
Tätig formally names the field or position in which someone works professionally. Eingespannt, on the other hand, is colloquial and describes someone being very busy with many tasks. Use „tätig“ to state someone’s profession or area of work and „eingespannt“ for their current workload.

Frau Neumann ist diese Woche wegen vieler Schichten stark eingespannt.
Ms. Neumann is very busy this week because of her many shifts.
Frau Neumann ist als Ärztin in einer Klinik tätig.
Ms. Neumann works as a doctor at a clinic.