brave / bold
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brave
The quality of overcoming fear or danger and acting decisively. It describes people or actions that show no fear even though the situation is difficult.
Es war sehr mutig von dir, ihm die Wahrheit zu sagen.
It was very brave of you to tell him the truth.
The adjective 'mutig' (brave) is derived directly from the noun 'der Mut' (courage). The ending '-ig' means 'having' or 'full of'. So someone who is mutig has courage.
Der Junge hatte den Mut, auf den hohen Baum zu klettern.
The boy had the courage to climb the tall tree.
To say who is acting bravely, the construction 'Es ist mutig von + [person in the dative]' is often used. It is usually followed by a subordinate clause with 'dass' or an infinitive with 'zu'.
Es ist mutig von ihm, seine Meinung offen zu sagen.
It is brave of him to state his opinion openly.
When 'mutig' (brave) describes a person or their action, it means 'fearless'. But when it refers to things like art, design, or a decision, it means 'unconventional' or 'daring'.
Die Feuerwehrleute waren sehr mutig.
The firefighters were very brave.
In the meaning of 'unconventional', 'mutig' (bold) is often combined with nouns such as 'Entscheidung' (decision), 'Design' (design), 'Farbe' (color), or 'Plan' (plan).
Das Unternehmen traf die mutige Entscheidung, in eine neue Technologie zu investieren.
The company made the bold decision to invest in a new technology.
afraid / anxious
'mutig' and 'bang' are opposites when it comes to fear. 'mutig' describes someone who acts despite fear or danger. 'bang' describes a feeling of fear or great worry. It is often used in expressions like 'mir ist bang' (I am afraid / uneasy) or 'ihm wird bang' (he becomes afraid / uneasy). So the difference is: 'mutig' emphasizes decisive action, while 'bang' emphasizes the anxious feeling.
Die Ärztin war mutig und blieb auch in der schwierigen Situation ruhig.
The doctor was brave and stayed calm even in the difficult situation.
Vor der Prüfung wurde Milan plötzlich bang.
Before the exam, Milan suddenly became afraid.