From a point to a plane, 2 inclined are drawn, forming angles of 30 and 45 with this plane

| August 5, 2021 | education

From a point to a plane, 2 inclined are drawn, forming angles of 30 and 45 with this plane. Find the angle between the bases of the inclined, if the projection of the smaller inclined is 3 cm, and the angle between the projections of the inclined is a straight line

Triangle ABD is rectangular and isosceles, since the angle BAD = 45, then BD = AB = 3 cm.

AD = 3 * √2 cm.

In a right-angled triangle ВСD, the leg ВD lies opposite the angle 30, then СД = ВД * 2 = 3 * 2 = 6 cm.Then ВС ^ 2 = СД ^ 2 – ВД ^ 2 = 36 – 9 = 27. ВС = 3 * √ 3 cm.

Since the angle between the projections is a straight line, the triangle ABC is rectangular, then AC ^ 2 = AB ^ 2 + BC ^ 2 = 9 + 27 = 36. AC = 6 cm.

In the triangle ACD, according to the cosine theorem: AC ^ 2 = AD ^ 2 + CD ^ 2 – 2 * AD * CD * CosD.

36 = 18 + 36 – 2 * 3 * √2 * 6 * Cos D.

36 * √2 * CosD = 18.

CosД = 18/36 * √2 = 1/2 * √2 = √2 / 4.

Then the angle D = arcsin (√2 / 4).

Answer: Angle D = arcsin (√2 / 4).



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