Question

Suppose each edge of the cube shown in the figure is 10 inches long. Find the sine and cosine of the angle formed by diagonals DE and DG.

Answer 1

Check the picture below.

`sin(EDG )=\cfrac{\stackrel{opposite}{10}}{\underset{hypotenuse}{10√(3)}}\implies sin(EDG )=\cfrac{1}{√(3)}\implies sin(EDG )=\cfrac{1}{√(3)}\cdot \cfrac{√(3)}{√(3)} \\\\\\ \stackrel{\textit{rationalizing the denominator}}{sin(EDG )=\cfrac{√(3)}{√(3^2)}\implies sin(EDG )=\cfrac{√(3)}{3}} \\\\[-0.35em] ~\dotfill`

`cos(EDG )=\cfrac{\stackrel{adjacent}{10√(2)}}{\underset{hypotenuse}{10√(3)}}\implies cos(EDG )=\cfrac{√(2)}{√(3)}\implies cos(EDG )=\cfrac{√(2)}{√(3)}\cdot \cfrac{√(3)}{√(3)} \\\\\\ \stackrel{\textit{rationalizing the denominator}}{cos(EDG )=\cfrac{√(6)}{√(3^2)}\implies cos(EDG )=\cfrac{√(6)}{3}}`