IDNLearn.com offers a reliable platform for finding accurate and timely answers. Get step-by-step guidance for all your technical questions from our dedicated community members.

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



Format the following question or task so that it is easier to read.
Fix any grammar or spelling errors.
Remove phrases that are not part of the question.
Do not remove or change LaTeX formatting.
Do not change or remove [tex] [/tex] tags.
If the question is nonsense, rewrite it so that it makes sense.
-----
[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]
-----

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]


Sagot :

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]