Here, we want to predict a linear dependent variable from circular independent variables. There are several ways to approach this. The main thing to check is whether the relation between your dependent variable (let's say $Y$) and the circular predictor (say $\theta$) has a sinusoidal shape. This is often the case, but not necessarily. Below is an example of data of this shape.
th <- rnorm(100, 1, 4) %% (2*pi)
err <- rnorm(100, mean = 0, sd = 0.8)
icp <- 10
bc <- 2
bs <- 3
y <- icp + bc * cos(th) + bs * sin(th) + err
If the data does have this shape, roughly, a good simple model for the data is then given by splitting the circular predictor $\theta$ up in a sine and a cosine component, and running a regular linear regression on these two components, in this case by:
lm(y ~ cos(th) + sin(th))
>lm(formula = y ~ cos(th) + sin(th))
>(Intercept) cos(th) sin(th)
> 10.12 2.04 2.95
Of course, this can be done for multiple predictors as well. A good introduction on this may be found in Pewsey, Neuhauser & Ruxton (2013), Circular Statistics in R.
As mentioned before, we may add terms as in a Fourier regression, but this can only be recommended if the relationship structurally exhibits very different forms, because higher-order Fourier regression introduces, IIRC, a large number of difficult to interpret parameters.