Exercises: sin, cos and tan in 3D

• a side with length $2\mathrm{cm}$ ,

• an area diagonal (belonging to the rectangle with side lengths 3 cm and 4 cm) (here named $f$).

• one of the spatial diagonals of the cuboid (here named $d$).

This triangle is right-angled.

The spatial diagonal of the cuboid in this triangle is the hypotenuse.

You can calculate the area diagonal $f$ using the Pythagorean theorem:

$f^2=(3\mathrm{cm}{)}^2+(4\mathrm{cm}{)}^2$

$f^2=9{\mathrm{cm}}^2+16{\mathrm{cm}}^2$

$f^2=25{\mathrm{cm}}^2$

$f=\sqrt{25{\mathrm{cm}}^2}$

$\Rightarrow f=5\mathrm{cm}$

Now that you know the length of $f$, you can calculate the angle $\alpha$ with the tangent:

$\tan \alpha =\frac{\mathrm{oppositecathetus}}{\mathrm{adjacentcathetus}}$

That means:

$\tan \alpha ={\displaystyle \frac{2\mathrm{cm}}{f}}$, so

$\tan \alpha ={\displaystyle \frac{2\mathrm{cm}}{5\mathrm{cm}}}={\displaystyle \frac{2}{5}}$