When to (and when not to) use the word "sind/bist"?
What are you doing? Was machst du?
warum bist du traurig? Why are you sad?
Why is the german word for "are" used sometimes, but not others? Meaning, are there any rules as to when are should be used or not used?
Thanks in advance!
The present progressive (and all other progressive tenses) is a form that does not exist in German. It is formed with "to be" as a helping word. It expresses that something is in progression, still ongoing. German uses other ways to express that. The easiest way is to neglect it. Often it is not important to differentiate between if it is still going on or if it happens now. So in your translation of the first example, you translate with simple present. You could be more precise and translate: "Was macht du gerade?" ("gerade" shows that it is still going on)
The other example is also in English in simple present, so the form of "to be" does not act as a helping word. Hence you have to translate it.
I hope I could be of help.
As explained in the Tips&Notes of Basics 1, German has no continuous aspect. There is no difference (in German) between, for example, I am swimming and I swim. In German, they are both ich schwimme. (You could think of it in the sense that German verbs tend to avoid touching each other at all costs, haha.)
Following this, in „Was machst du (gerade)?" bist has effectively pulled off a dazzling disappearing act, as die deutsche Sprache has no need for bist as an assisting verb. In „Warum bist du traurig?" bist (a declension of sein or to be) is a complete verb.
The same thing goes for sind/seid/bin/ist. (The other declensions of sein.)
Basically, in English "I am eating" and "I eat" mean two different things. "I am eating" means that I am presently eating, whereas "I eat" simply means that sometimes I eat, whether I eat daily, right now, or tonight.
In German, however, there is no distinction between "I am eating" and "I eat." They are both, "Ich esse." One would never say, "Ich bin essen."
As for the second, you need the "are" in there. There aren't any other verbs. If you omitted "bist," you'd basically be saying, "Why sad you?" That isn't grammatically correct in either language.
This is basically what I've been taught in my deutsche Klasse! I hope it helps you out.