Angle between a point and $z$-axis in the view where $x = y$

Question

I am trying to find the angle between a point and the $z$-axis in the view where $x=y$.

For example: a point $C = (-24, -24, -118)$ as illustrated in the next picture.

C in 3D space

how to find the angle shown in the picture below:

The angle needed

PS. My background is not mathematics. An example would be very much appreciated.

Answer 1

Let use dot product by vectors $\vec u=OC=(-24,-24,-118)$ and $\vec v=(0,0,1)$ that is

$$\cos \theta=\frac{\vec u\cdot \vec v}{|\vec u||\vec v|}\implies \theta = \arccos \frac{\vec u\cdot \vec v}{|\vec u||\vec v|}$$

Note that in this way we find the angle measured from positive direction of z axes. The angle indicated in the picture is its complement $180°-\theta$.

Answer 2

Hint:

Imagine you draw a perpendicular line to $xy$-plane from point $C$. If that line meet $xy$-plane at $D$, then the angle you are looking for is $\angle DOC$. and $(OD)^2=(-x)^2+(-y)^2=2x^2$. Now you can find $\tan (\angle DOC)$.