How to solve for when this trigonometric function intersects the line $y=1$?

Question

How can I solve for $\alpha$ in $$4\sin\left(\frac{\alpha}{2}\right)\cos\left(\frac{\alpha}{2}\right)^{3}\left(t-r\right)+\sin\left(\frac{\alpha}{2}\right)=1$$ on the domain $0\leq\alpha\leq\pi$? Clearly, one solution is when $\alpha=\pi$, but through plotting, it seems to only hold true when $t-r$ is less than a value around $0.3$. When $t-r$ is greater than this value, it seems to have different solutions.

NSolve[HoldForm[4*Sin[\[Alpha]/2]*Cos*(\[Alpha]/2)^3*(t - r) + Sin[\[Alpha]/2]] == 1 && 0 <= \[Alpha] <= Pi, \[Alpha], Reals]

I am very new so I don't have much code.

Answer 1

sol = Solve[
     4*Sin[a/2]*Cos[a/2]^3*tr + Sin[a/2] == 1 && 0 <= a <= Pi, {a, tr}, 
      Reals]

(*   {{a -> \[Pi]}, {tr -> 
       ConditionalExpression[1/4 Csc[a/2] Sec[a/2]^3 (1 - Sin[a/2]), 
        0 < a < \[Pi]]}}   *)

Plot[Evaluate[tr /. sol], {a, 0, Pi}, PlotRange -> {0, 20}, 
     GridLines -> Automatic]

{min = Minimize[tr /. sol[[2]], a], min // N}

(*   {{-(1/4) Csc[
2 ArcTan[Root[1 + 2 #1 - 10 #1^2 + 2 #1^3 + #1^4 &, 3]]] Sec[
2 ArcTan[Root[1 + 2 #1 - 10 #1^2 + 2 #1^3 + #1^4 &, 3]]]^3 (-1 + 
 Sin[2 ArcTan[
    Root[1 + 2 #1 - 10 #1^2 + 2 #1^3 + #1^4 &, 3]]]), {a -> 
4 ArcTan[
  Root[1 + 2 #1 - 10 #1^2 + 2 #1^3 + #1^4 &, 
   3]]}}, {0.287482, {a -> 1.75015}}}   *)

Two solutions: 1. for a==Pi, any tr you like 2. For 0 < a < Pi, tr depending on a as shown in the graph.