Vehicular ad-hoc networks (VANETs) consist of highly mobile and self-organized nodes that wirelessly communicate with one another and transmit a variety of information, some of which may be critical information. Due to the rapidly changing topology of VANETs and the high-speed mobility of the participating vehicles, the routing of packets in these networks is a challenging task. Moreover, the presence of malicious vehicles in VANETs makes the correct routing of packets more challenging. In this paper, we propose a trust-based geographic routing protocol for VANETs (TGRV) that limits the participation of malicious vehicles in routing. In TGRV, to select the next-hop, a sender not only considers the distance, speed, and direction of its neighbors but also evaluates their direct trust and recommendation trust. For this purpose, TGRV uses a monitoring system that allows each vehicle to monitor the correct packet forwarding rate of its next-hop. In this way, the vehicle updates its direct trust to the next-hop and retransmits the packets if the packets are lost. The vehicle also sends its observations of the next-hop to its neighbors with a push-based notification, and based on that, the neighbors can update their recommendation trust about the next-hop. The monitoring system applies distance prediction in a modified promiscuous mode to better estimate the correct packet forwarding rate of the next-hop. To enhance the accuracy of trust management, the trust values of interactions are reduced over time by using a decay factor. TGRV uses the number and trust of two-hop neighbors, which helps in selecting the next-hop that is in a more trusted zone. Our extensive simulations on OMNeT++ show that when the percentage of malicious vehicles in an urban scenario increases from 0 to 25%, the packet delivery ratio of TGRV decreases from 95.1 to 88.7%, which performs very well compared to the GPSR and PGRP protocols. Also, the end-to-end delay of TGRV increases from 3.79 to 7.07 s and the average hop count of TGRV increases from 6.7 to 9.9, which are significant increases compared to the other two protocols. These increases are acceptable because, in the other two protocols, there is no monitoring and retransmission, and the probability of packet delivery failure on longer routes is higher. The cost and security analysis also show that the memory, communication, and computational overheads of TGRV are acceptable, and TGRV is resistant and resilient to some important trust-based attacks.