Sure, let's find a standard angle that is coterminal with [tex]\(\theta = -\frac{14\pi}{3}\)[/tex] and then evaluate [tex]\(\cot \theta\)[/tex] using the unit circle.
1. Understanding Coterminal Angles:
- An angle is coterminal with another angle if their difference is a multiple of [tex]\(2\pi\)[/tex]. In other words, adding or subtracting [tex]\(2\pi\)[/tex] (or [tex]\(360^\circ\)[/tex]) repeatedly to an angle will result in an angle that falls within a standard range, usually [tex]\([0, 2\pi)\)[/tex] for radians.
2. Convert [tex]\(\theta\)[/tex] to a Standard Angle:
- We need to bring [tex]\(\theta = -\frac{14\pi}{3}\)[/tex] within the range [tex]\([0, 2\pi)\)[/tex].
- To do this, we add [tex]\(2\pi\)[/tex] repeatedly until we get a positive coterminal angle in the desired range.
- Mathematically, this can be found using the modulo operation:
[tex]\[
\text{coterminal angle} = -\frac{14\pi}{3} \mod 2\pi
\][/tex]
Since [tex]\(2\pi\)[/tex] is equivalent to [tex]\(\frac{6\pi}{3}\)[/tex], we have:
[tex]\[
\text{coterminal angle} = -\frac{14\pi}{3} \mod \frac{6\pi}{3}
\][/tex]
3. Finding the Coterminal Angle:
- Calculate:
[tex]\[
-\frac{14\pi}{3} + 2\pi \times k
\][/tex]
where [tex]\(k\)[/tex] is an integer such that the result is in the range [tex]\([0, 2\pi)\)[/tex].
- By computation, we find:
[tex]\[
-\frac{14\pi}{3} + \frac{6\pi}{3} \times \left(\left\lfloor \frac{14}{6} \right\rfloor + 1\right)
\][/tex]
Here,
[tex]\[
\left\lfloor \frac{14}{6} \right\rfloor = 2 \Rightarrow \text{Next integer} = 3
\][/tex]
Thus:
[tex]\[
-\frac{14\pi}{3} + \frac{6\pi}{3} \times 3 = -\frac{14\pi}{3} + \frac{18\pi}{3} = \frac{4\pi}{3}
\][/tex]
- The standard coterminal angle for [tex]\(\theta\)[/tex] is:
[tex]\[
\frac{4\pi}{3}
\][/tex]
4. Evaluate [tex]\(\cot \theta\)[/tex] Using the Unit Circle:
- The unit circle value for [tex]\(\theta = \frac{4\pi}{3}\)[/tex]:
[tex]\(\frac{4\pi}{3}\)[/tex] lies in the third quadrant where both sine and cosine are negative.
- The reference angle for [tex]\(\frac{4\pi}{3}\)[/tex] is:
[tex]\[
\pi - \left(\frac{4\pi}{3} - \pi\right) = \frac{\pi}{3}
\][/tex]
- [tex]\(\cot \theta\)[/tex] is:
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
where [tex]\(\theta = \frac{4\pi}{3}\)[/tex], being coterminal with [tex]\(\frac{\pi + \pi/3 = 4\pi/3}\)[/tex], has coordinates:
[tex]\[
(\cos (\frac{4\pi}{3}), \sin (\frac{4\pi}{3})) = \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)
\][/tex]
Thus:
[tex]\[
\cot (\frac{4\pi}{3}) = \frac{\cos (\frac{4\pi}{3})}{\sin (\frac{4\pi}{3})} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\][/tex]
Therefore:
- The coterminal angle is [tex]\( \frac{4\pi}{3} \)[/tex].
- The value of [tex]\(\cot \theta\)[/tex] is approximately [tex]\(0.5773502691896252\)[/tex].
The numerical values:
[tex]\[
(-14.660765716752367, 4.188790204786391, 0.5773502691896252)
\][/tex]
match our computed result, confirming that [tex]\( \cot( \frac{4\pi}{3}) \approx 0.5773502691896252 \)[/tex].
