Why isn't $\tan \left(\frac{\theta}{4} \right) \times 4$ the same as $\tan \left(\frac{\theta}{2} \right) \times 2$?

Question

I am trying to calculate the length of an opposite side of a triangle (the blue line in the picture below)

I know that $\theta = 60\deg$ and the green line is $y=3$ so to calculate the blue line, i.e. $x$:

$tan(\frac {\theta} {2}) = \frac {1} {2}x / y$

alternatively,

$y \cdot tan(\frac {\theta} {2}) = \frac 1 {2}x$ so $x = 2 \cdot y \cdot tan(\frac {\theta} {2})$

So $x = 2 \cdot 3 \cdot tan(30)$

So $ x = 2 \cdot 3 \cdot 0.57735056839 =$ 3.46410341035

Now, let's say I wanted to do the same thing with cutting the pyramid into fourths. I think I should get the same answer, but my results are different.

Here I would think we would use:

$tan(\frac {\theta} {4}) = \frac {1} {4}x / y$

because instead of just halving $\theta$ and $x$ we cut them in fourths so:

So $x = 4 \cdot 3 \cdot tan(15)$

So $ x = 4 \cdot 3 \cdot 0.26794877678 =$ 3.21538532136

Why are these results slightly different? Please explain. Thank you

Answer 1

As Martin R points out in the comments, tangent is not a linear function. Essentially (after adjustment by a factor of $4$), you are asking why

$$ \tan 2x \not= 2\tan x $$

In the diagram above, the red line segment represents the tangent of the red angle, and the combined red and blue segments represent the tangent of the combined red and blue angles. You can see that the blue angle contributes much more to the tangent, because it is "higher up" on the circle.