"I thought that you were at home."
Translation:Jag trodde att du var hemma.
In English, you can use "Believe" and "Think" interchangeably in this sentence, but in Swedish you can't.
Swedish has three verbs that correspond to three of the meanings of English "Think", these are:
"Tro" which means to think that something is true or false. i.e. Believing.
"Tycka" which means to have an opinion about something. "Jag tycker att det ser fint ut" - "I think it looks nice".
And "Tänka", which means to think about something. It can also mean "Intend", such as in the phrase "Jag tänker inte göra det" - "I don't intend on doing that"
Lundgren8 wrote about the difference between tycka, tänka, tro here: https://www.duolingo.com/comment/5577824
In this specific case, I'd say the thing is that thought in often used in sentences like this in English meaning more or less believe. I mean, when you say I thought you were at home, you're really saying that you 'held that belief', rather than 'that thought crossed your mind'.
Edit: cross posting. I agree with what Berniebud says, but I'm leaving my comment too.