앞에서 이미 두 분이 잘 설명해 주셨네요.

1. 설명해 주신 내용을 정리하면 다음과 같습니다.

뜨다 : informal present form
떴다 : informal past form

뜹니다 : formal present form
떴습니다 : formal past form

2. 제 견해를 말씀드립니다. (이상한 점이 있으면 지적하여 주십시오.)

그러나, (설명하기 힘들지만) '떴다' 혹은 '떴습니다'는 현재형 혹은 현재분사형으로 쓰입니다.
즉, "무지개가 하늘에 아름답게 떴습니다."는 " ... 떠 있습니다."라는 의미이지 "(과거에) ... 떠 있었습니다."라는 의미가 아닙니다.

Therefore, "무지개가 하늘에 아름답게 떴습니다." is translated as "There is a beautiful rainbow in the sky," or "A beautiful rainbow has appeared in the sky." However, the sentence is translated neither "A beautiful rainbow is appearing in the sky," nor ""There was a beautiful rainbow in the sky."

I believe that 떳습니다 in your sentence is spelled incorrectly.  The infinitive form of the verb is 뜨다 which means to rise, come up,  or float. 

Your sentence indicates past tense. As 뜨다 is irregular it should be conjugated just like 쓰다 (to write).

Informal past form: 떴다

Formal past form: 떴습니다. Which is how your sentence should end. 

*If I am wrong, I certainly welcome corrections.

Shou님, that is a very good point.  Although it is written in a past form it has a meaning of "has risen/ has appeared and is there" so your English translation is perfect.  Either, "There is a beautiful rainbow in the sky," or "A beautiful rainbow has appeared in the sky."

(I am writing in English for Hannah, but am glad that you wrote in Korean as it is easier for me to follow Korean grammar when the information is in Korean.  :)  

It means to 'rise' in the past form. 

(correction)

'떴다' 혹은 '떴습니다'는 현재형 혹은 과거완료형으로 쓰입니다.