Nationally representative surveys track public opinion, yet they ask only a limited set of questions each year, limiting its potential to capture historical changes. To fill this gap, we develop a large language model (LLM)-based framework for predicting missing responses in repeated cross-sectional surveys by incorporating embeddings for questions, respondents, and survey periods. We introduce two new applications of LLMs to survey research: retrodiction (predicting year-level missing opinions) and unasked opinion prediction (predicting entirely missing opinions). Using data from the 1972-2021 General Social Surveys, our LLM-based models perform strongly in retrodicting masked GSS opinions through cross-validation and public opinions measured by other organizations in years when the GSS did not ask them. These capabilities enable us to recover missing trends and pinpoint when public attitudes changed, such as the rising support for same-sex marriage. However, performance remains modest for unasked opinion prediction. We show when our models outperform established benchmarks, examine which opinions and and respondents are more predictable, and evaluate whether our approach reduces LLMs' tendency to homogenize predicted responses. Our study demonstrates that LLMs and surveys can mutually enhance each other: LLMs broaden survey potential, while surveys calibrate LLMs for simulating human opinions.