Answer :
To determine \(\cos\left(\frac{5\pi}{3}\right)\), let's proceed step-by-step.
1. Understand the given angle:
[tex]\[ \frac{5\pi}{3} \][/tex]
This is an angle in radians.
2. Converting the angle to a more familiar form:
The angle \(\frac{5\pi}{3}\) is greater than \(2\pi\). To find a corresponding angle within the standard \(0\) to \(2\pi\) interval, we can subtract \(2\pi\) (which is \(6\pi/3\)) from it:
[tex]\[ \frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3} \][/tex]
Since cosine is an even function, \(\cos(-x) = \cos(x)\), we have:
[tex]\[ \cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) \][/tex]
3. Identify the cosine value of the angle \(\frac{\pi}{3}\):
From the unit circle, the cosine of \(\frac{\pi}{3}\) is a well-known value:
[tex]\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
4. Verify and compare with given options:
Upon calculation, the closest value to \(\cos\left(\frac{5\pi}{3}\right)\) is indeed:
[tex]\[ 0.5000000000000001 \approx \frac{1}{2} \][/tex]
Let's verify each given option:
- A. \(-\frac{\sqrt{2}}{2}\) is approximately \(-0.707\), which is significantly different from \(0.5\).
- B. \(\frac{\sqrt{2}}{2}\) is approximately \(0.707\), which is also different from \(0.5\).
- C. \(\frac{1}{2}\) matches our result exactly.
- D. \(\frac{\sqrt{3}}{2}\) is approximately \(0.866\), which does not match our result.
Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{2}} \][/tex]
1. Understand the given angle:
[tex]\[ \frac{5\pi}{3} \][/tex]
This is an angle in radians.
2. Converting the angle to a more familiar form:
The angle \(\frac{5\pi}{3}\) is greater than \(2\pi\). To find a corresponding angle within the standard \(0\) to \(2\pi\) interval, we can subtract \(2\pi\) (which is \(6\pi/3\)) from it:
[tex]\[ \frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3} \][/tex]
Since cosine is an even function, \(\cos(-x) = \cos(x)\), we have:
[tex]\[ \cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) \][/tex]
3. Identify the cosine value of the angle \(\frac{\pi}{3}\):
From the unit circle, the cosine of \(\frac{\pi}{3}\) is a well-known value:
[tex]\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
4. Verify and compare with given options:
Upon calculation, the closest value to \(\cos\left(\frac{5\pi}{3}\right)\) is indeed:
[tex]\[ 0.5000000000000001 \approx \frac{1}{2} \][/tex]
Let's verify each given option:
- A. \(-\frac{\sqrt{2}}{2}\) is approximately \(-0.707\), which is significantly different from \(0.5\).
- B. \(\frac{\sqrt{2}}{2}\) is approximately \(0.707\), which is also different from \(0.5\).
- C. \(\frac{1}{2}\) matches our result exactly.
- D. \(\frac{\sqrt{3}}{2}\) is approximately \(0.866\), which does not match our result.
Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{2}} \][/tex]
