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Completion requirements
Suppose \(\theta\) is an angle in standard position with terminal
arm in Quadrant III. Determine the exact values of \(\rm{sin} \thinspace
\theta\) and \(\rm{tan} \thinspace \theta\), given \(\rm{cos}
\thinspace \theta = -\frac{2}{5}\).
\(\begin{align}
r^2 &= x^2 + y^2 \\
\left( 5 \right)^2 &= \left( { - 2} \right)^2 + y^2 \\
25 &= 4 + y^2 \\
21 &= y^2 \\
\pm \sqrt {21} &= y \\
\end{align} \)
Because this angle terminates in Quadrant III, \(y = -\sqrt{21}\).
\(\begin{align}
\sin \theta &= \frac{y}{r} \\
\sin \theta &= \frac{{ - \sqrt {21} }}{5} \\
\end{align}\)
\(\begin{align}
\tan \theta &= \frac{y}{x} \\
\tan \theta &= \frac{{ - \sqrt {21} }}{{ - 2}} \\
\tan \theta &= \frac{{\sqrt {21} }}{2} \\
\end{align}\)
\(\begin{align}
r^2 &= x^2 + y^2 \\
\left( 5 \right)^2 &= \left( { - 2} \right)^2 + y^2 \\
25 &= 4 + y^2 \\
21 &= y^2 \\
\pm \sqrt {21} &= y \\
\end{align} \)
Because this angle terminates in Quadrant III, \(y = -\sqrt{21}\).
\(\begin{align}
\sin \theta &= \frac{y}{r} \\
\sin \theta &= \frac{{ - \sqrt {21} }}{5} \\
\end{align}\)
\(\begin{align}
\tan \theta &= \frac{y}{x} \\
\tan \theta &= \frac{{ - \sqrt {21} }}{{ - 2}} \\
\tan \theta &= \frac{{\sqrt {21} }}{2} \\
\end{align}\)