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Determine which of the following is equivalent to the given expression:

[tex]\[
\cos \left(\frac{2 \pi}{3}\right)
\][/tex]

A. [tex]\(\sin \left(\frac{2 \pi}{3}\right)\)[/tex]

B. [tex]\(\sec \left(\frac{\pi}{2}\right)\)[/tex]

C. [tex]\(\sec \left(\frac{-\pi}{6}\right)\)[/tex]

D. [tex]\(\sin \left(\frac{-\pi}{6}\right)\)[/tex]


Sagot :

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].