Final answer:
The modal verb 'should' in the given sentence is used to make a prediction, indicating that it is likely the sun will come out by noon based on the weather forecast information.
Step-by-step explanation:
In the sentence 'I checked the forecast, and the sun should come out by noon,' the bold modal verb is 'should.' This modal verb is used to express a prediction or reasonable expectation about future events, based on what the speaker knows, such as a weather forecast. The use of 'should' here does not indicate certainty, but it suggests a strong likelihood that the condition (the sunshine) will occur. Modal verbs like 'should' are an element of verb mood, which is used to convey different levels of necessity, possibility, obligation, or permission.