Let $$ \sin \theta = s, \cos \theta = c $$
We can use Lagrange Multiplier method to maximize/minimize
$$ c+s + 2 s c $$
subject to trig constraint
$$ c^2+s^2 =1 $$
$$\dfrac{1+2 s }{1+2c}=\dfrac{2c}{2s}$$
which simplifies to
$$ (s-c) (1+2s+2c)=0\;$$
$$s=c,\; s+c=-\dfrac12$$
These are two conditions one each for maximum and minimum evaluation:
First case maximum
$$ s=c; s^2+c^2=1\rightarrow s=c=1/\sqrt{2};\;$$
Maximum value
$$ 1/\sqrt{2}+ 1/\sqrt{2}+1 = 1+\sqrt{2}$$
Second case minimum
$$ s+c=-\frac12,\; s^2+c^2=1\;$$
$$ s= \dfrac{\sqrt7-1}{4};\;c= -\dfrac{\sqrt7+1}{4};$$
Minimum value
$$ s+c+ 2sc = =\dfrac{-5}{4}. $$
Both solutions verify on plot of $f( \theta)= s+c+sc\;$.