Alright, let's determine which of the given trigonometric expressions is equivalent to [tex]\(\cos \left(\frac{2 \pi}{3}\right)\)[/tex].
First, let's recall that:
[tex]\[
\cos \left(\frac{2 \pi}{3}\right)
\][/tex]
is the cosine of an angle that is in the second quadrant of the unit circle (since [tex]\( \frac{2\pi}{3} \)[/tex] is between [tex]\( \pi/2 \)[/tex] and [tex]\( \pi \)[/tex]). The cosine of an angle in the second quadrant is always negative. Specifically, [tex]\(\frac{2 \pi}{3} = \pi - \frac{\pi}{3}\)[/tex], so we can use the identity for cosine:
[tex]\[
\cos (\pi - \theta) = -\cos (\theta)
\][/tex]
For [tex]\(\theta = \frac{\pi}{3}\)[/tex]:
[tex]\[
\cos \left(\frac{2 \pi}{3}\right) = -\cos \left(\frac{\pi}{3}\right)
\][/tex]
And knowing that [tex]\(\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}\)[/tex]:
[tex]\[
\cos \left(\frac{2 \pi}{3}\right) = -\frac{1}{2}
\][/tex]
So, we know:
[tex]\[
\cos \left(\frac{2 \pi}{3}\right) = -0.5
\][/tex]
Now, let's evaluate each of the given options:
1. [tex]\(\sin \left(\frac{2 \pi}{3}\right)\)[/tex]:
- Since [tex]\(\sin(\pi - \theta) = \sin(\theta)\)[/tex], we have:
[tex]\(
\sin \left(\frac{2 \pi}{3}\right) = \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \approx 0.866
\)[/tex]
2. [tex]\(\sec \left(\frac{\pi}{2}\right)\)[/tex]:
- [tex]\(\sec(\theta) = \frac{1}{\cos(\theta)}\)[/tex]. For [tex]\(\theta = \frac{\pi}{2}\)[/tex]:
[tex]\(
\cos\left(\frac{\pi}{2}\right) = 0
\)[/tex]
which implies
[tex]\(
\sec \left(\frac{\pi}{2}\right) = \frac{1}{0}
\)[/tex]
which is undefined and tends towards infinity.
3. [tex]\(\sec \left(\frac{-\pi}{6}\right)\)[/tex]:
- Similar to the above, [tex]\(\frac{1}{\cos(\theta)}\)[/tex]. For [tex]\(\theta = -\frac{\pi}{6}\)[/tex]:
[tex]\(
\cos \left( -\frac{\pi}{6} \right) = \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2}
\)[/tex]
thus
[tex]\(
\sec \left( -\frac{\pi}{6} \right) = \frac{1}{\cos \left( \frac{\pi}{6} \right)} = \frac{2}{\sqrt{3}} \approx 1.155
\)[/tex]
4. [tex]\(\sin \left(\frac{-\pi}{6}\right)\)[/tex]:
- We know [tex]\(\sin(-\theta) = -\sin(\theta)\)[/tex], so:
[tex]\(
\sin \left( -\frac{\pi}{6} \right) = -\sin \left( \frac{\pi}{6} \right) = -\frac{1}{2} = -0.5
\)[/tex]
Comparing each of these:
[tex]\[
\cos \left(\frac{2 \pi}{3}\right) = -0.5
\][/tex]
matches:
4. [tex]\(\sin \left(\frac{-\pi}{6}\right) = -0.5\)[/tex]
Thus, the equivalent expression to [tex]\(\cos \left(\frac{2 \pi}{3}\right)\)[/tex] is:
[tex]\(\boxed{\sin \left(\frac{-\pi}{6}\right)}\)[/tex].
