Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
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 appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
