Write tan(cos^-1(- x/5)) as an algebraic Expression in terms of X. ?

Question

$tan ({cos}^{- 1} (- \frac{x}{5}))$ $Tan(Cos^-1 (- x/5))$

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Answer 1 · 2017-04-05T21:02:45

$tan ({cos}^{- 1} (- \frac{x}{5})) = \pm \frac{\sqrt{25 - x^{2}}}{x}$ $tan(cos^-1 (- x/5)) = pm (sqrt(25 - x^2))/(x)$

You want:

$tan ({cos}^{- 1} (- \frac{x}{5})) = tan α$ $tan(cos^-1 (- x/5)) color(blue)(= tan alpha)$

Well:

$α = {cos}^{- 1} (- \frac{x}{5})$ $alpha = cos^-1 (- x/5)$

$\Rightarrow cos α = - \frac{x}{5}$ $implies cosalpha = - x/5$

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So in the first instance, this is:

$tan α = \frac{\sqrt{25 - x^{2}}}{- x}$ $tan alpha = (sqrt(25 - x^2))/(-x)$

The other possibility is:

Velazquez

$tan α = \frac{- \sqrt{25 - x^{2}}}{- x} = \frac{\sqrt{25 - x^{2}}}{x}$ $tan alpha = (- sqrt(25 - x^2))/(-x) = (sqrt(25 - x^2))/(x)$

So:

$tan ({cos}^{- 1} (- \frac{x}{5})) = \pm \frac{\sqrt{25 - x^{2}}}{x}$ $tan(cos^-1 (- x/5)) = pm (sqrt(25 - x^2))/(x)$