Carole Yarbrough

2021-12-20

Write the trigonometric expression as an algebraic expression in u.
$\mathrm{sec}\left({\mathrm{cos}}^{-1}u\right)$

Bertha Jordan

Expert

Inside of the braces can be rewritten
$\mathrm{cos}\theta =u$ and from Pythagoras theorem, we have another side as $\sqrt{1-{u}^{2}}$
$\frac{1}{\mathrm{sec}\theta }=u$
$\mathrm{sec}\theta =\frac{1}{u}$
Answer: $\mathrm{sec}\theta =\frac{1}{u}$

Medicim6

Expert

$\mathrm{arccos}\left(u\right)=\theta$ say
$\mathrm{cos}\left(\theta \right)=u$
$\mathrm{sec}\left(\theta \right)=\frac{1}{u}$
$\theta =arc\mathrm{sec}\left(\frac{1}{u}\right)$
$\mathrm{sec}\left(\mathrm{arccos}\left(u\right)\right)$
$\mathrm{sec}\left(arc\mathrm{sec}\left(\frac{1}{u}\right)\right)$
Answer: $=\frac{1}{u}$

RizerMix

Expert

$\mathrm{sec}\left({\mathrm{cos}}^{-1}u\right)$
let ${\mathrm{cos}}^{-1}u=x$
$⇒\mathrm{cos}x=u$
$⇒\frac{1}{\mathrm{sec}x}=u$
$⇒\mathrm{sec}x=\frac{1}{u}$
we have ${\mathrm{cos}}^{-1}u=x$
$⇒\mathrm{sec}\left({\mathrm{cos}}^{-1}u\right)=\frac{1}{x}$

Do you have a similar question?