Trigonometric question combined with questions about Mathematica

Question

I'm new to Mathematica and I am struggling with the Syntax and the functions of this "programming" language.
I'm looking for a way to solve sin2 α cos3 α = tan4 α
But if I try to run it through the Solve[] function I recieve an error.
I'm not looking for hand holding and having someone give me the answer. What I am trying to do is to learn which functions work where and how I successfully can calculate questions like these in Mathematica.

Bob Hanlon · Accepted Answer

Clear["Global`*"]

eqn = Sin[2 a] Cos[3 a] == Tan[4 a];

The functions are periodic so there are many periodic solutions.

Solve[eqn, a] // Short[#, 4] &

Restrict the range of a to some region of interest

sol = Solve[{eqn, 0 <= a <= 2 Pi}, a]

The approximate numeric values are

sol // N

(* {{a -> 0.}, {a -> 1.5708}, {a -> 3.14159}, {a -> 4.71239}, {a -> 
   6.28319}, {a -> 5.60235}, {a -> 0.680831}, {a -> 4.11126}, {a -> 2.17193}} *)

Graphically,

Plot[Evaluate[List @@ eqn], {a, 0, 2 Pi},
 Epilog -> {Red, AbsolutePointSize[6], Point[{a, eqn[[1]]} /. sol]}, 
 PlotLegends -> "Expressions"]

Trigonometric question combined with questions about Mathematica

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