1) If it is cyclic, as in 365 day in a year, then a nonlinear regression using a cosine function should work.

2) If it is dispersal of an insect placed at the center of an arena, then I would look in the literature to find relevant citations. I know that this sort of thing has been published, though the analysis may be too simple to be useful for you.

3) If the problem is slope of a landslide. Are you sure it is not normally distributed? I haven't seen too many slopes at 284 degrees. I would bet that the angles are all between zero and 90 degrees. A slope of zero will not generate much of a landslide, and the only stable "slope at 90 degrees is solid rock. The rock could collapse, but that is not exactly a landslide. So maybe 30 to 70 degrees? Within that range, do you have a normal distribution? If so, there is no problem here, use ANOVA.

It depends a bit on how you are using the data. So I suspect that as slope increases the probability of a landslide increases. The cumulative probability follows an S shaped curve that reaches 1 at 80 degrees. You could consider logistic regression or a logit transformation for the probability of a landslide. However, if I look at the distribution of slopes at which landslides occur I should see an increase up to some value, and then a decrease. The decrease occurs because there will be very few cases where I observe a landslide at 78 degrees because it is unlikely that the soil will hold sufficiently to achieve this slope. The distribution of slopes might be normally distributed.

However, I am not a geologist. There may be a feature of the data that I have not considered. There may also be a standard method in the geology literature that I am entirely unaware of.

In the end we came up with a data transformation considering that the angles of the slopes need to be, as you said, between zero and 90 degrees.

Thank you so much for the answer, especially for the different cases explanation.