Solve for [tex]\( x \)[/tex]:
[tex]\[ 3x = 6x - 2 \][/tex]



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[tex]$
\cos \left(\frac{5 \pi}{3}\right)=
$[/tex]
[tex]$\qquad$[/tex]
A. [tex]$\frac{\sqrt{2}}{2}$[/tex]
B. [tex]$-\frac{\sqrt{2}}{2}$[/tex]
C. [tex]$\frac{1}{2}$[/tex]
D. [tex]$\frac{\sqrt{3}}{2}$[/tex]
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Response:
[tex]\[ \cos \left(\frac{5 \pi}{3}\right) = \][/tex]

A. [tex]\(\frac{\sqrt{2}}{2}\)[/tex]

B. [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]

C. [tex]\(\frac{1}{2}\)[/tex]

D. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]



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]