What's the point with neural networks if you can only predict linear test data?
Neural networks can generalize and successfully predict outside their training data. This ability is hindered by overfitting, where the network memorizes the training data and does not perform well on unseen data.
As with any other problem faced with neural networks, it is key for the network to have an inductive bias that is appropriate for the problem data. From this SO answer:

Every machine learning algorithm with any ability to generalize beyond the training data that it sees has some type of inductive bias, which are the assumptions made by the model to learn the target function and to generalize beyond training data.

For instance, convolutional networks perform well in image data due to their spatial locality inductive bias.
The main problem in your examples is that the function you are modeling is periodic while the activation functions you are using lack a periodic inductive bias. This problem is studied in the article Neural Networks Fail to Learn Periodic Functions and How to Fix It presented at NeurIPS 2020. From the abstract:

[...] we prove and demonstrate experimentally that the standard
activations functions, such as ReLU, tanh, sigmoid, along with their variants, all fail to learn to extrapolate simple periodic functions. We hypothesize that this is due to their lack of a “periodic” inductive bias. As a fix of this problem, we propose a new activation, namely, $x + sin^2 (x)$, which achieves the desired periodic inductive bias to learn a periodic function while maintaining a favorable optimization property of the ReLU-based activations.