If sin (π / 6 + α) = 3 / 5 α ∈ (π / 3,5 π / 6), find Tan (5 π / 12 + α)

If sin (π / 6 + α) = 3 / 5 α ∈ (π / 3,5 π / 6), find Tan (5 π / 12 + α)

∵sin(π/6+α)=3/5,α∈(π/3,5π/6) ,∴π/6+α∈(π/2,π),∴cos(π/6+α)=√1-[sin(π/6+α)]^2=√(1-9/25)=-4/5,∴tan(π/6+α)=sin(π/6+α)/cos(π/6+α)=-3/4,tan(5π/12+α)=tan[(π/6+α)+π/4]=[tan(π/6...

Judge whether there are acute angles x, y satisfying x + 2Y = 2 π / 3 and Tan (x / 2) * tany = 2 - √ 3. If there are, calculate these two acute angles
I think this way: because x + 2Y = 2 π / 3, so x / 2 + y = π / 3
So tan (x / 2) + tany = Tan (x / 2 + y) * [1 - (Tan (x / 2) * tany)] = 3 - √ 3

tan(x/2)+tany=3-√3
tan(x/2)tany=2-√3
Simultaneous solutions of two equations

Solving the equation sin2x = SiNx

2sinxcosx=sinx
cosx=1/2
X = 60 degrees or pi / 3
(of course, the range of X is limited to 0 to 90 degrees.)