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

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

• If you divide the red angle into four equal parts then the corresponding segments on the blue line do not have all the same length. – Martin R May 30 at 7:25
• @MartinR how would I make them all have equal lengths? – Startec May 30 at 7:26
• You can't have both: four equal angles and four equal length. The simple reason is that the tangent is not a linear function. – Martin R May 30 at 7:27
• Your second figure does not divide the blue triangle into right triangles, and $\tan \theta = \frac {\text{opposite}}{\text{adjacent}}$ only applies to right triangles. – Doug M May 30 at 7:50
• @DougM the middle two triangles would still be right triangles. – Startec May 30 at 8:12

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