To determine why [tex]\(\cos \frac{2 \pi}{3} \neq \cos \frac{5 \pi}{3}\)[/tex], let's analyze the given angles and their respective cosine values step-by-step.
1. Convert the Angles to Decimal Degrees:
- [tex]\(\frac{2 \pi}{3}\)[/tex] radians is equivalent to [tex]\(120^\circ\)[/tex].
- [tex]\(\frac{5 \pi}{3}\)[/tex] radians is equivalent to [tex]\(300^\circ\)[/tex].
2. Determine the Quadrants:
- An angle of [tex]\(120^\circ\)[/tex] (or [tex]\(\frac{2 \pi}{3}\)[/tex]) is located in the second quadrant.
- An angle of [tex]\(300^\circ\)[/tex] (or [tex]\(\frac{5 \pi}{3}\)[/tex]) is located in the fourth quadrant.
3. Characteristics of Cosine in Different Quadrants:
- In the second quadrant, the cosine of an angle is negative.
- In the fourth quadrant, the cosine of an angle is positive.
4. Calculate or Use Known Values of Cosines:
- [tex]\(\cos \frac{2\pi}{3} \approx -0.5\)[/tex].
- [tex]\(\cos \frac{5\pi}{3} \approx 0.5\)[/tex].
5. Compare the Cosine Values:
- These values confirm that [tex]\(\cos \frac{2\pi}{3}\)[/tex] is negative and [tex]\(\cos \frac{5\pi}{3}\)[/tex] is positive.
Now, given the information and the properties of cosine in various quadrants:
- The correct explanation is: Cosine is negative in the second quadrant and positive in the fourth quadrant.
The specific cosine values are:
- [tex]\(\cos \frac{2 \pi}{3} \approx -0.5\)[/tex].
- [tex]\(\cos \frac{5 \pi}{3} \approx 0.5\)[/tex].
Thus, through the characteristics of the cosine function in different quadrants, we see that [tex]\(\cos \frac{2 \pi}{3} \neq \cos \frac{5 \pi}{3}\)[/tex] because cosine takes different signs in the second and fourth quadrants.
