 # Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function. Given: displaystyle tan{{left({{cos}^{ -{{1}}}{5}}{x}right)}}=? Wribreeminsl 2021-02-25 Answered
Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function.
Given:
$\mathrm{tan}\left({\mathrm{cos}}^{-1}5x\right)=?$
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Step 1
Let,
$a={\mathrm{cos}}^{-1}5x$
Then, $\mathrm{cos}a=5x$
Therefore,
${\mathrm{sin}}^{2}a+{\mathrm{cos}}^{2}a=1$
${\mathrm{sin}}^{2}a=1-{\mathrm{cos}}^{2}a$
$\mathrm{sin}a=±\sqrt{1-{\mathrm{cos}}^{2}a}$
$\mathrm{sin}a=±\sqrt{1-{\left(5x\right)}^{2}}$
$\mathrm{sin}a=±\sqrt{1-25{x}^{2}}$
Step 2
Hence,
$\mathrm{tan}\left({\mathrm{cos}}^{-1}5x\right)=\mathrm{tan}a$
$=\frac{\mathrm{sin}a}{\mathrm{cos}a}$
$\mathrm{tan}\left({\mathrm{cos}}^{-1}5x\right)=±\frac{\sqrt{1-25{x}^{2}}}{5x}$
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