With the rapidly increasing number of cloud-based services, selecting a service provider is becoming more and more difficult. Among the many factors to be considered, trust in a given service and in a service provider is often critical. Appraisal of trust is a complex process, information about the offered service's quality needs to be collected from a number of sources, while user requirements may place different emphasis on the various quality indicators. Several frameworks have been proposed to facilitate service provider selection, however, only very few of them are built on existing cloud standards, and adaptability to different contexts is still a challenge. This paper proposes a new trust framework, called Context-Aware Multifaceted Trust Framework (CAMFT), to assist in evaluating trust in cloud service providers. CAMTF is flexible and context aware: it considers trust factors, users and services. When making recommendations, CAMFT employs a combination of mathematical methods that depend on the type of trust factors, and it takes both service characteristics and user perspective into account. A case study is also presented to demonstrate CAMFT's applicability to practical cases.