Question: Given all variables are in $\mathbb{R}$, and $\{ a, d\} \neq 0$, solve for $\theta$:

$$a \sin^2 \theta + b \sin \theta + c + d \cos^2 \theta + e \cos \theta + f = 0$$

as posed in [this question](https://math.stackexchange.com/questions/4577312/solving-quadratic-function-of-sine-and-cosine#4577312).

The straightforward approach does not work:

```
Assuming[a != 0 \[And] b != 0,
 Solve[a Sin[\[Theta]]^2 + b Sin[\[Theta]] + c + d Cos[\[Theta]]^2 + 
    e Cos[\[Theta]] + f == 0, \[Theta]]]

```

One can force "smart" substitutions, as given in the solution to the linked source problem. Is there a direct way to get the solution using *Mathematica*?

Answer: 
Mathematica has no trouble solving

```
eqn = a Sin[θ]^2 + b Sin[θ] + c + d Cos[θ]^2 + 
    e Cos[θ] + f == 0;

subs = {Sin[θ] -> sin, Cos[θ] -> cos};

solution = Solve[(eqn /. subs) && sin^2 + cos^2 == 1, {sin, cos}];

```

though the solution is rather large, and may not be useful to you.
I think Weirstaß-substitution isn't "awkward" and gives a clear direct solution:

Transforming the equation using Weirstaß-substitution `\[Theta] -> 2 ArcTan[u\[Theta]]` gives

```
eqW = a Sin[\[Theta]]^2 + b Sin[\[Theta]] + c + d Cos[\[Theta]]^2 + 
e Cos[\[Theta]] + f /. \[Theta] -> 2 ArcTan[u\[Theta]] // TrigExpand // Collect[#, {a, b, c, d, e, f}, Simplify] &     

```

$$
\frac{4 a \text{u$\theta $}^2}{\left(\text{u$\theta$}^2+1\right)^2}+\frac{2 b \text{u$\theta $}}{\text{u$\theta$}^2+1}+c+\frac{d \left(\text{u$\theta$}^2-1\right)^2}{\left(\text{u$\theta$}^2+1\right)^2}+\frac{e \left(1-\text{u$\theta$}^2\right)}{\text{u$\theta $}^2+1}+f
$$

Mathematica

```
Solve[eqW == 0, u\[Theta], Reals] /. u\[Theta] -> Tan[\[Theta]/2];

```

finds four solutions in the form of Root objects

`{Tan[\[Theta]/2] ->Root[c + d + e + f + 2 b #1 + (4 a + 2 c - 2 d + 2 f) #1^2 + 2 b #1^3 + (c + d - e + f) #1^4 &, 1]}`

Solutions are lengthy.