Does an angle become twice larger in a right triangle if the correspondence 1:3 becomes 1:1.5? (tangent)

Asked Jan 23 '23 at 12:55

Active Jan 23 '23 at 13:51

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Does an angle become twice larger in a right triangle if the (tangent) correspondence $1:3$ becomes $1:1.5$?

I do in fact know that it isn't true. $$2\tan^{-1}\left(\frac{1}{3}\right) \neq \tan^{-1}\left(\frac{1}{1.5}\right)$$ But I don't know why and would like to know.

edited Jan 23 '23 at 13:51

Blue

75,673

asked Jan 23 '23 at 12:55

Flowlessrusher

3

beacause $\tan x$ is not simply a multiple of $x$ (the graph is not a straight line) – Hagen von Eitzen Jan 23 '23 at 13:04
It's basically the same as asking if $\tan x$ is a linear function. It isn't. For really small values of $x$ you can sometimes approximate $\tan x \approx x$ ($x$ in radians) but in general this isn't true. – Matti P. Jan 23 '23 at 13:42
Yeah I know that but how can I explain that tan x isn't a linear function – Flowlessrusher Jan 23 '23 at 13:55
Related (duplicate?): "Why is the tangent of 22.5 degrees not 1/2?" See, in particular, my answer. ... When you get down to it, the whole reason trigonometry exists as a subject is to bridge the very different behaviors between side-lengths and angle-measures in a triangle. – Blue Jan 23 '23 at 14:00
I read you answer but I am having trouble understanding it. – Flowlessrusher Jan 23 '23 at 14:22

Does an angle become twice larger in a right triangle if the correspondence 1:3 becomes 1:1.5? (tangent)

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