Primary Trigonometric Ratios of Quadrantal Angles
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Primary Trigonometric Ratios of Quadrantal Angles
The quadrantal angles are angles in standard position where the terminal arm lies on one of the axes. There are five quadrantal angles possible in one rotation \(0^\circ\), \(90^\circ\), \(180^\circ\), \(270^\circ\), and \(360^\circ\). As with any angle in standard position, the trigonometric ratios can be determined.
For \(0^\circ\), the terminal arm lies on the positive \(x\)-axis.
Suppose the point \((x, 0)\) is a point on the positive \(x\)-axis.
Then, \(x = x\), \(y = 0\), and \(r = x\) (the length of \(x\)).
\(\begin{align}
\sin \theta &= \frac{y}{r} \\
\sin 0^\circ &= \frac{0}{x} \\
\sin 0^\circ &= 0 \\
\end{align}\)
\sin \theta &= \frac{y}{r} \\
\sin 0^\circ &= \frac{0}{x} \\
\sin 0^\circ &= 0 \\
\end{align}\)
\(\begin{align}
\cos \theta &= \frac{x}{r} \\
\cos 0^\circ &= \frac{x}{x} \\
\cos 0^\circ &= 1 \\
\end{align}\)
\cos \theta &= \frac{x}{r} \\
\cos 0^\circ &= \frac{x}{x} \\
\cos 0^\circ &= 1 \\
\end{align}\)
\(\begin{align}
\tan \theta &= \frac{y}{x} \\
\tan 0^\circ &= \frac{0}{x} \\
\tan 0^\circ &= 0 \\
\end{align}\)
\tan \theta &= \frac{y}{x} \\
\tan 0^\circ &= \frac{0}{x} \\
\tan 0^\circ &= 0 \\
\end{align}\)
For \(90^\circ\), the terminal arm lies on the positive \(y\)-axis.
Suppose the point \((0, y)\) is a point on the positive \(y\)-axis.
Then, \(x = 0\), \(y = y\), and \(r = y\) (the length of \(y\)).
\(\begin{align}
\sin \theta &= \frac{y}{r} \\
\sin 90^\circ &= \frac{y}{y} \\
\sin 90^\circ &= 1 \\
\end{align}\)
\sin \theta &= \frac{y}{r} \\
\sin 90^\circ &= \frac{y}{y} \\
\sin 90^\circ &= 1 \\
\end{align}\)
\(\begin{align}
\cos \theta &= \frac{x}{r} \\
\cos 90^\circ &= \frac{0}{y} \\
\cos 90^\circ &= 0 \\
\end{align}\)
\cos \theta &= \frac{x}{r} \\
\cos 90^\circ &= \frac{0}{y} \\
\cos 90^\circ &= 0 \\
\end{align}\)
\(\begin{align}
\tan \theta &= \frac{y}{x} \\
\tan 90^\circ &= \frac{y}{0} \\
\tan 90^\circ &= {\rm{undefined}} \\
\end{align}\)
\tan \theta &= \frac{y}{x} \\
\tan 90^\circ &= \frac{y}{0} \\
\tan 90^\circ &= {\rm{undefined}} \\
\end{align}\)
For \(180^\circ\), the terminal arm lies on the negative \(x\)-axis.
Suppose the point \((-x, 0)\) is a point on the negative \(x\)-axis.
Then, \(x = -x\), \(y = 0\), and \(r = x\) (the length of \(x\)).
\(\begin{align}
\sin \theta &= \frac{y}{r} \\
\sin 180^\circ &= \frac{0}{x} \\
\sin 180^\circ &= 0 \\
\end{align}\)
\sin \theta &= \frac{y}{r} \\
\sin 180^\circ &= \frac{0}{x} \\
\sin 180^\circ &= 0 \\
\end{align}\)
\(\begin{align}
\cos \theta &= \frac{x}{r} \\
\cos 180^\circ &= \frac{{ - x}}{x} \\
\cos 180^\circ &= -1 \\
\end{align}\)
\cos \theta &= \frac{x}{r} \\
\cos 180^\circ &= \frac{{ - x}}{x} \\
\cos 180^\circ &= -1 \\
\end{align}\)
\(\begin{align}
\tan \theta &= \frac{y}{x} \\
\tan 180^\circ &= \frac{0}{{ - x}} \\
\tan 180^\circ &= 0 \\
\end{align}\)
\tan \theta &= \frac{y}{x} \\
\tan 180^\circ &= \frac{0}{{ - x}} \\
\tan 180^\circ &= 0 \\
\end{align}\)
For \(270^\circ\), the terminal arm lies on the negative \(y\)-axis.
Suppose the point \((0, -y)\) is a point on the negative \(y\)-axis.
Then, \(x = 0\), \(y = -y\), and \(r = y\) (the length of \(y\)).
\(\begin{align}
\sin \theta &= \frac{y}{r} \\
\sin 270^\circ &= \frac{{ - y}}{y} \\
\sin 270^\circ &= -1 \\
\end{align}\)
\sin \theta &= \frac{y}{r} \\
\sin 270^\circ &= \frac{{ - y}}{y} \\
\sin 270^\circ &= -1 \\
\end{align}\)
\(\begin{align}
\cos \theta &= \frac{x}{r} \\
\cos 270^\circ &= \frac{0}{y} \\
\cos 270^\circ &= 0 \\
\end{align}\)
\cos \theta &= \frac{x}{r} \\
\cos 270^\circ &= \frac{0}{y} \\
\cos 270^\circ &= 0 \\
\end{align}\)
\(\begin{align}
\tan \theta &= \frac{y}{x} \\
\tan 270^\circ &= \frac{{ - y}}{0} \\
\tan 270^\circ &= {\rm{undefined}} \\
\end{align}\)
\tan \theta &= \frac{y}{x} \\
\tan 270^\circ &= \frac{{ - y}}{0} \\
\tan 270^\circ &= {\rm{undefined}} \\
\end{align}\)
For \(360^\circ\), the terminal arm lies on the positive \(x\)-axis.
The ratios will be exactly the same as for \(0^\circ\). Suppose the point \((x, 0)\) is a point on the positive \(x\)-axis. Then, \(x = x\), \(y = 0\), and \(r = x\) (the length of \(x\)).
\(\begin{align}
\sin \theta &= \frac{y}{r} \\
\sin 360^\circ &= \frac{0}{x} \\
\sin 360^\circ &= 0 \\
\end{align}\)
\sin \theta &= \frac{y}{r} \\
\sin 360^\circ &= \frac{0}{x} \\
\sin 360^\circ &= 0 \\
\end{align}\)
\(\begin{align}
\cos \theta &= \frac{x}{r} \\
\cos 360^\circ &= \frac{x}{x} \\
\cos 360^\circ &= 1 \\
\end{align}\)
\cos \theta &= \frac{x}{r} \\
\cos 360^\circ &= \frac{x}{x} \\
\cos 360^\circ &= 1 \\
\end{align}\)
\(\begin{align}
\tan \theta &= \frac{y}{x} \\
\tan 360^\circ &= \frac{0}{x} \\
\tan 360^\circ &= 0 \\
\end{align}\)
\tan \theta &= \frac{y}{x} \\
\tan 360^\circ &= \frac{0}{x} \\
\tan 360^\circ &= 0 \\
\end{align}\)
