Compare Your Answers
Completion requirements
Determine the primary trigonometric ratios of \(\theta\), if the
point \((7, -5)\) lies on the terminal arm of angle \(\theta\) in
standard position.
Sketch the angle and reference triangle, and then determine the length of the hypotenuse.
\(\begin{align}
r &= \sqrt {x^2 + y^2} \\
r &= \sqrt {\left( 7 \right)^2 + \left( -5 \right)^2} \\
r &= \sqrt {49 + 25} \\
r &= \sqrt {74} \\
\end{align}\)
The primary trigonometric ratios of \(\theta\) are:
\(\begin{align}
r &= \sqrt {x^2 + y^2} \\
r &= \sqrt {\left( 7 \right)^2 + \left( -5 \right)^2} \\
r &= \sqrt {49 + 25} \\
r &= \sqrt {74} \\
\end{align}\)
The primary trigonometric ratios of \(\theta\) are:
\(\begin{align}
\sin \theta &= \frac{y}{r} \\
\sin \theta &= \frac{{ - 5}}{{\sqrt {74} }} \cdot \frac{{\sqrt {74} }}{{\sqrt {74} }} \\
\sin \theta &= -\frac{{5\sqrt {74} }}{{74}} \\
\end{align}\)
\sin \theta &= \frac{y}{r} \\
\sin \theta &= \frac{{ - 5}}{{\sqrt {74} }} \cdot \frac{{\sqrt {74} }}{{\sqrt {74} }} \\
\sin \theta &= -\frac{{5\sqrt {74} }}{{74}} \\
\end{align}\)
\(\begin{align}
\cos \theta &= \frac{x}{r} \\
\cos \theta &= \frac{7}{{\sqrt {74} }} \cdot \frac{{\sqrt {74} }}{{\sqrt {74} }} \\
\cos \theta &= \frac{{7\sqrt {74} }}{{74}} \\
\end{align}\)
\cos \theta &= \frac{x}{r} \\
\cos \theta &= \frac{7}{{\sqrt {74} }} \cdot \frac{{\sqrt {74} }}{{\sqrt {74} }} \\
\cos \theta &= \frac{{7\sqrt {74} }}{{74}} \\
\end{align}\)
\(\begin{align}
\tan \theta &= \frac{y}{x} \\
\tan \theta &= \frac{{ - 5}}{7} \\
\tan \theta &= - \frac{5}{7} \\
\end{align}\)
\tan \theta &= \frac{y}{x} \\
\tan \theta &= \frac{{ - 5}}{7} \\
\tan \theta &= - \frac{5}{7} \\
\end{align}\)