Answer :
To determine the value of [tex]\(\cos \left(\frac{5 \pi}{3}\right)\)[/tex], let's follow a step-by-step approach.
1. Understand the Angle in the Unit Circle:
- The given angle is [tex]\(\frac{5\pi}{3}\)[/tex] radians. To understand where this lies on the unit circle, we can convert it to degrees if needed, but it's not necessary here since radians are sufficient.
2. Locate the Angle:
- [tex]\(\frac{5\pi}{3}\)[/tex] radians can also be analyzed as a position on the unit circle. Recall that [tex]\(2\pi\)[/tex] radians is a full circle, so:
[tex]\[ 2\pi = \frac{6\pi}{3} \][/tex]
Hence, [tex]\(\frac{5\pi}{3}\)[/tex] is slightly less than a full circle (by [tex]\(\frac{\pi}{3}\)[/tex] radians or 60 degrees).
3. Reference Angle:
- To determine the cosine value, find the reference angle. The reference angle for [tex]\(\frac{5\pi}{3}\)[/tex] is calculated as:
[tex]\[ 2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3} \][/tex]
- Therefore, the reference angle is [tex]\(\frac{\pi}{3}\)[/tex].
4. Using Known Values:
- The cosine of [tex]\(\frac{\pi}{3}\)[/tex] is a known value:
[tex]\[ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
5. Consider the Quadrant:
- Since [tex]\(\frac{5\pi}{3}\)[/tex] is in the fourth quadrant (where [tex]\(2\pi - \theta\)[/tex] angles lie), and in the fourth quadrant, the cosine function is positive.
6. Conclusion:
- Given the reference angle [tex]\(\frac{\pi}{3}\)[/tex] and the positive cosine in the fourth quadrant, we conclude:
[tex]\[ \cos \left(\frac{5\pi}{3}\right) = \frac{1}{2} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{2}} \][/tex]
1. Understand the Angle in the Unit Circle:
- The given angle is [tex]\(\frac{5\pi}{3}\)[/tex] radians. To understand where this lies on the unit circle, we can convert it to degrees if needed, but it's not necessary here since radians are sufficient.
2. Locate the Angle:
- [tex]\(\frac{5\pi}{3}\)[/tex] radians can also be analyzed as a position on the unit circle. Recall that [tex]\(2\pi\)[/tex] radians is a full circle, so:
[tex]\[ 2\pi = \frac{6\pi}{3} \][/tex]
Hence, [tex]\(\frac{5\pi}{3}\)[/tex] is slightly less than a full circle (by [tex]\(\frac{\pi}{3}\)[/tex] radians or 60 degrees).
3. Reference Angle:
- To determine the cosine value, find the reference angle. The reference angle for [tex]\(\frac{5\pi}{3}\)[/tex] is calculated as:
[tex]\[ 2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3} \][/tex]
- Therefore, the reference angle is [tex]\(\frac{\pi}{3}\)[/tex].
4. Using Known Values:
- The cosine of [tex]\(\frac{\pi}{3}\)[/tex] is a known value:
[tex]\[ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
5. Consider the Quadrant:
- Since [tex]\(\frac{5\pi}{3}\)[/tex] is in the fourth quadrant (where [tex]\(2\pi - \theta\)[/tex] angles lie), and in the fourth quadrant, the cosine function is positive.
6. Conclusion:
- Given the reference angle [tex]\(\frac{\pi}{3}\)[/tex] and the positive cosine in the fourth quadrant, we conclude:
[tex]\[ \cos \left(\frac{5\pi}{3}\right) = \frac{1}{2} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{2}} \][/tex]
