Question

Evaluate tan ( cos^-1 ( 1/8 ) ) , giving your answer as an exact value (no decimals)

Answer 1

`\bf cos^(-1)\left( \cfrac{1}{8} \right)=\theta \qquad \qquad cos(\theta )=\cfrac{\stackrel{adjacent}{1}}{\stackrel{hypotenuse}{8}}\impliedby \textit{let's find the \underline{opposite}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies √(c^2-a^2)=b \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ \pm√(8^2-1^1)=b\implies \pm√(63)=b ~\hfill tan(\theta )=\cfrac{\stackrel{opposite}{\pm√(63)}}{\stackrel{adjacent}{1}} \\\\\\ ~\hspace{34em}`

`\bf \stackrel{\textit{keeping in mind that}}{tan\left(cos^(-1)\left( (1)/(8) \right) \right)}\implies tan(\theta )`