Savanah Boone

2022-07-01

Given the cube ABCD. EFGH. The point M is on the edge AD such that $|AM|=2|MD|$ . Calculate the tangent of the angle between the planes BCF and BGM.
A) $3\sqrt{2}$
B) $2\sqrt{2}$
C) $\frac{3}{2}\sqrt{2}$
D) $\frac{1}{2}\sqrt{2}$
E) $\sqrt{2}$

sniokd

Expert

Step 1
The value that you've got, $\frac{\sqrt{22}}{11}$ is the cosine of the angle. Then
$\mathrm{tan}\alpha =\frac{\mathrm{sin}\alpha }{\mathrm{cos}\alpha }=\frac{\sqrt{1-{\mathrm{cos}}^{2}\alpha }}{\mathrm{cos}\alpha }\phantom{\rule{0ex}{0ex}}=\frac{\sqrt{1-\frac{22}{{11}^{2}}}}{\frac{\sqrt{22}}{11}}=\sqrt{\frac{99}{22}}=3\frac{1}{\sqrt{2}}=\frac{3}{2}\sqrt{2}$

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