Question

Find the exact value of the expression. Do not use a calculator. If tan θ = 9, find the exact value of cot θ

Answer 1

`\bf tan(\theta )=9\implies tan(\theta )=\cfrac{\stackrel{opposite}{9}}{\stackrel{adjacent}{1}}\impliedby \textit{let's find the \underline{hypotenuse}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=√(a^2+b^2) \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases}`

`\bf c=√(1^2+9^2)\implies c=√(82) \\\\[-0.35em] ~\dotfill\\\\ cos(\theta )=\cfrac{\stackrel{adjacent}{1}}{\stackrel{hypotenuse}{√(82)}}\implies \stackrel{\textit{rationalizing the denominator}}{cos(\theta )=\cfrac{1}{√(82)}\cdot \cfrac{√(82)}{√(82)}}\implies cos(\theta )=\cfrac{√(82)}{82}`