How to prove that (tan b / tab a) > (b/a) whenever 0 < a < b < π/2?

Question

How to prove that (tan b / tab a) > (b/a) whenever 0 < a < b < π/2?

I am able to show that tan x > x for 0 < x < π/2 but this doesn't seems to get me through to the final conclusion I'm looking for.

1 Answer

Impact of this question

2036 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-08-07T23:42:46

Consider the function:

$f (x) = \frac{tan x}{x}$ $f(x) = tanx/x$ in the interval $(0, \frac{π}{2})$ $(0,pi/2)$

we have that:

$\frac{d f}{d x} = \frac{d}{d x} (\frac{tan x}{x}) = \frac{\frac{x}{{cos}^{2} x} - tan x}{x^{2}} = \frac{x - sin x cos x}{x^{2} {cos}^{2} x}$ $(df)/dx = d/dx (tanx/x) = (x/cos^2x-tanx)/x^2 = (x-sinxcosx)/(x^2cos^2x)$

Now, as $x > sin x$ $x > sinx$ and $cos x < 1$ $cosx < 1$ , then we have that:

$x - sin x cos x > 0$ $x - sinx cosx > 0$

and

$\frac{d}{d x} (\frac{tan x}{x}) > 0$ $d/dx (tanx/x) > 0$

so that the function is strictly increasing and:

$b > a \Rightarrow \frac{tan b}{b} > \frac{tan a}{a}$ $b > a => tanb/b > tana/a$

As for $0 < a < b < \frac{π}{2}$ $0 < a < b < pi/2$ we have that $b > 0$ $b > 0$ and $tan a > 0$ $tana > 0$ this is equivalent to:

$\frac{tan b}{tan a} > \frac{b}{a}$ $tanb/tana > b/a$

Science

Math

Humanities

... and beyond

How to prove that (tan b / tab a) > (b/a) whenever 0 < a < b < π/2?

I am able to show that tan x > x for 0 < x < π/2 but this doesn't seems to get me through to the final conclusion I'm looking for.

1 Answer

Related questions

Impact of this question