Show that if 0<Ө<π/2 then sinӨ tanӨ> 2(1-cosӨ).

Asked on by roshan-rox

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embizze's profile pic

embizze | High School Teacher | (Level 2) Educator Emeritus

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Show `sinthetatanthetagt2(1-costheta)` if `0<theta<pi/2` :

You cannot assume that the inequality holds, so you cannot work across the inequality. (e.g. you cannot add to both sides, etc... This is assuming that the inequality is valid.)

`sinthetatantheta=sintheta(sintheta)/(costheta)=(sin^2theta)/costheta`  Use the Pythagoren relationship


`=(1+costheta)/costheta (1-costheta)`

Consider `(1+costheta)/costheta=1/costheta+1`

On `0<theta<pi/2` we have `0<costheta<1 ==> 0<1<1/costheta`

Thus `1/costheta+1>1+1=2`

Therefore `sinthetatantheta=(1/costheta+1)(1-costheta)>2(1-costheta)` as required.

aruv's profile pic

aruv | High School Teacher | (Level 2) Valedictorian

Posted on











Which is always true. Thus

`sin(theta)tan(theta)>2(1-cos(theta))`  is true.

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