Example 3
Completion requirements
| Example 3 |
Multimedia |
The second ring from the centre of the Aztec Calendar marks the days of
the Aztec month, which number \(20\). Determine which days of the month
would give a ratio of tan \(\theta = -1.3764\).
Step 1: Determine in which quadrants the solutions will lie.
Recall that tan \(\theta\) is negative in Quadrants II and IV.
Step 2: Determine the reference angle.
\(\begin{align}
\tan \theta &= 1.3764 \\
\theta &= \tan ^{ - 1} \left( {1.3764} \right) \\
\theta &= 54.000...^\circ \\
\end{align}\)
Step 3: Determine the angles in standard position.
Quadrant II:
\(\begin{align}
\theta &= 180^\circ - \theta _R \\
\theta &= 180^\circ - 54.000^\circ \\
\theta &= 126.000^\circ \\
\theta &\doteq 126^\circ \\
\end{align}\)
Quadrant IV:
\(\begin{align}
\theta &= 360^\circ - \theta _R \\
\theta &= 360^\circ - 54.000^\circ \\
\theta &= 306.000^\circ \\
\theta &\doteq 306^\circ \\
\end{align}\)
Step 4: Determine the days of the month that correspond to the angles.
Set up a proportion.
The two days of the month that have a tangent ratio of \(–1.3764\) are day 7 and day 17.
Recall that tan \(\theta\) is negative in Quadrants II and IV.
Step 2: Determine the reference angle.
\(\begin{align}
\tan \theta &= 1.3764 \\
\theta &= \tan ^{ - 1} \left( {1.3764} \right) \\
\theta &= 54.000...^\circ \\
\end{align}\)
Step 3: Determine the angles in standard position.
Quadrant II:
\(\begin{align}
\theta &= 180^\circ - \theta _R \\
\theta &= 180^\circ - 54.000^\circ \\
\theta &= 126.000^\circ \\
\theta &\doteq 126^\circ \\
\end{align}\)
Quadrant IV:
\(\begin{align}
\theta &= 360^\circ - \theta _R \\
\theta &= 360^\circ - 54.000^\circ \\
\theta &= 306.000^\circ \\
\theta &\doteq 306^\circ \\
\end{align}\)
Step 4: Determine the days of the month that correspond to the angles.
Set up a proportion.
\(\begin{align}
\frac{x \thinspace {\rm {days}}}{{20}} &= \frac{{126^\circ }}{{360^\circ }} \\
x &= 7\thinspace {\rm days} \\
\end{align}\)
\frac{x \thinspace {\rm {days}}}{{20}} &= \frac{{126^\circ }}{{360^\circ }} \\
x &= 7\thinspace {\rm days} \\
\end{align}\)
\(\begin{align}
\frac{x \thinspace {\rm {days}}}{{20}} &= \frac{{306^\circ }}{{360^\circ }} \\
x &= 17\thinspace {\rm days} \\
\end{align}\)
\frac{x \thinspace {\rm {days}}}{{20}} &= \frac{{306^\circ }}{{360^\circ }} \\
x &= 17\thinspace {\rm days} \\
\end{align}\)
The two days of the month that have a tangent ratio of \(–1.3764\) are day 7 and day 17.