At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
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].
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].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
