In the triangular prism abc-a1b1c1, the length of each edge is equal, find the angle between the straight line CB1 and the plane aa1bb1

In the triangular prism abc-a1b1c1, the length of each edge is equal, find the angle between the straight line CB1 and the plane aa1bb1

Take the midpoint o of AB and connect Co, b1o,
It can be proved that the angle between the straight line CB1 and the plane aa1bb1 is the size of ∠ cob1
From the cosine theorem, we can get ∠ cob1 = arctg (radical 3 / radical 5)

If the volume of the regular triangular prism abc-a1b1c1 is 12 pieces of 3 and the side length of the bottom surface is 4, then the tangent value of the angle between the straight line A1B and the bottom surface ABC

Regular triangular prism abc-a1b1c1
A1A is vertical to the bottom, and the angle formed by A1B and the bottom is called angle a1ba, which is in plane a1abb1
If the side length of the bottom surface is 4, the area is 4 root number 3
A1A＝3
The tangent value is A1A, ab = 3 / 4

All edge lengths of straight triangular prism abc-a1b1c1 are a, and D is the midpoint of edge Bici. Find the tangent of the angle between AD and bottom ABC

In plane cc1b1b, make dd1 perpendicular to CB through point D, the perpendicular foot is D1, and connect AD1. Because in straight triangular prism abc-a1b1c1, so: dd1 perpendicular to plane ABC. So: AD1 is the projection of ad in plane ABC. So: angle Dad1 is the angle of ad in plane ABC, because straight triangular prism

It is known that the side edges and the bottom sides of the triangular prism abc-a1b1c1 are equal, and the projection of A1 in the ground ABC is the midpoint of BC, then the cosine of the angle between AB and CC1 is?

Let the midpoint of BC be d
CC1//AA1
So the angle between AB and CC1 = the angle between AB and Aa1
Let Aa1 = AB = BC = CA = 2A
BD=DC=a
AD^2+DC^2=AC^2
AD=sqrt(3)a
AD^2+A1D^2=AA1^2
A1D=a
A1D^2+DB^2=A1B^2
A1B^2=2a
A1b=sqrt(2)a
In triangle aa1b
AA1=AB=2a
A1B=sqrt(2)a
From cosine theorem
A1B ^ 2 = Aa1 ^ 2 + AB ^ 2-2aa1 * AB * cos (included angle)
2 = 8-8cos (included angle)
The angle COS is 3 / 4