A balanced dataset is generally beneficial to underwater acoustic target recognition. However, the imbalanced class distribution is always meted out in a real scene. To address this, a weighted cross entropy loss function based on trigonometric function is proposed. Then, the proposed loss function is applied in a multi-scale residual convolutional neural network (named MR-CNN-A network) embedded with an attention mechanism for the recognition task. Firstly, a multi-scale convolution kernel is used to obtain multi-scale features. Then, an attention mechanism is used to fuse these multi-scale feature maps. Furthermore, a cosx-function-weighted cross-entropy loss function is used to deal with the class imbalance in underwater acoustic data. This function adjusts the loss ratio of each sample by adjusting the loss interval of every mini-batch based on cosx term to achieve a balanced total loss for each class. Two imbalanced underwater acoustic data sets, ShipsEar and autonomous underwater vehicle (self-collected data) are used to evaluate the proposed network. The experimental results show that the proposed network outperforms the support vector machine and a simple convolutional neural network. Compared with the other three loss functions, the proposed loss function achieves better stability and adaptability. The results strongly demonstrate the validity of the proposed loss function and the network.