Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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.
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.
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.