Trust is more like you allow or believe something without fear. Believe is you think something is the truth. Therefore, trust implies negative consequences and believe does not. Also you can trust something without believing it.
"I trust that the sun will set at 7:04 pm," would be incorrect. Why would you fear the exact time the sun sets? Will something negative occur if it does not set at 7:04 pm?
"I believe that the sun will set at 7:04 pm," would be correct. You probably watched the Weather Channel and they told you what time the sun would set, or something like that.
"I trust my car will get me to work every day," would be correct. Whether or not the statement comes true, I choose not to buy a new car that I can rely on. I may not even believe the statement. Perhaps I simply can't afford a new car. If my car does not perform, I will most likely face negative consequences at work.
"I believe my car will get me to work every day," means something slightly different. I have no need for a new car, since I believe that my current one is reliable.