Answer:
30°
Step-by-step explanation:
You want the dihedral angle formed by an equilateral triangle and plane P, given one leg of the triangle lies in plane P and the other legs form an angle α with the plane such that sin(α) = (√3)/4.
Diagram
The attachment shows a diagram of this geometry. The equilateral triangle is ∆ABC, with segment AB in the z-plane (plane P). We have made AB=BC=AC=1, so sin(α) is the height of point C above plane P, (√3)/4.
As in any equilateral triangle, the altitude of the triangle, OC, is (√3)/2 times the side length. This make ∆OCP a right triangle with hypotenuse (√3)/2 and leg CP = (√3)/4.
Dihedral angle POC is ...
[tex]\angle POC=\arcsin\left(\dfrac{(\dfrac{\sqrt{3}}{4})}{(\dfrac{\sqrt{3}}{2})}\right)=\arcsin\left(\dfrac{2}{4}\right)=30^\circ[/tex]
The dihedral angle between the plane of the triangle and plane P is 30°.
