Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Let's go through Henry's steps and identify where he made errors in his process of finding the exact value of [tex]\( \cos \frac{10 \pi}{3} \)[/tex].
### Henry's Steps:
1. Subtract [tex]\( 2 \pi \)[/tex] from [tex]\( \frac{10 \pi}{3} \)[/tex] as many times as possible:
[tex]\[ \frac{10 \pi}{3} - 2 \pi = \frac{10 \pi}{3} - \frac{6 \pi}{3} = \frac{4 \pi}{3} \][/tex]
This step is correct. [tex]\( \frac{10 \pi}{3} \)[/tex] reduced by cycle of [tex]\( 2 \pi \)[/tex] lands at [tex]\( \frac{4 \pi}{3} \)[/tex].
2. Find the reference angle for [tex]\( \frac{4 \pi}{3} \)[/tex]:
Here, Henry states:
[tex]\[ \frac{3 \pi}{2} - \frac{4 \pi}{3} = \frac{\pi}{6} \][/tex]
However, this approach is incorrect. The correct way to find the reference angle for [tex]\( \frac{4 \pi}{3} \)[/tex] is to subtract [tex]\( \pi \)[/tex]:
[tex]\[ \text{Reference angle} = \frac{4 \pi}{3} - \pi = \frac{4 \pi}{3} - \frac{3 \pi}{3} = \frac{\pi}{3} \][/tex]
3. The cosine value for [tex]\( \frac{\pi}{6} \)[/tex] is [tex]\( \frac{\sqrt{3}}{2} \)[/tex]:
While this cosine value is correct for [tex]\( \frac{\pi}{6} \)[/tex], the reference angle [tex]\( \frac{\pi}{6} \)[/tex] is wrong.
4. The cosine value is positive because [tex]\( \frac{\pi}{6} \)[/tex] is in the first quadrant:
This conclusion is also incorrect because [tex]\( \frac{4 \pi}{3} \)[/tex] is located in the third quadrant where cosine values are negative.
### Correct Procedure:
Let's correct the solution step-by-step:
1. Reduce the angle:
[tex]\[ \frac{10 \pi}{3} \equiv \frac{4 \pi}{3} \, (\text{mod} \, 2 \pi) \][/tex]
This reduction is correct.
2. Determine the quadrant:
[tex]\( \frac{4 \pi}{3} \)[/tex] is in the third quadrant of the unit circle.
3. Find the reference angle:
[tex]\[ \text{Reference angle} = \frac{4 \pi}{3} - \pi = \frac{\pi}{3} \][/tex]
4. Evaluate the cosine for the reference angle in the third quadrant:
[tex]\[ \cos \frac{\pi}{3} = \frac{1}{2} \][/tex]
Since cosine is negative in the third quadrant:
[tex]\[ \cos \frac{4 \pi}{3} = -\frac{1}{2} \][/tex]
Therefore, the correct value of [tex]\( \cos \frac{4 \pi}{3} \)[/tex] is [tex]\( -\frac{1}{2} \)[/tex].
### Identifying Henry's Errors:
- Henry incorrectly calculated the reference angle. Instead of using [tex]\( \frac{4 \pi}{3} - \pi \)[/tex], he incorrectly used [tex]\( \frac{3 \pi}{2} \)[/tex].
- He misunderstood the quadrant properties for cosine. In the third quadrant, cosine values should be negative.
### Correct Answer:
- Henry’s reference angle calculation is incorrect.
- Henry’s determination of the cosine sign is incorrect.
The correct value of [tex]\( \cos \frac{10 \pi}{3} \)[/tex] is [tex]\( -\frac{1}{2} \)[/tex].
Thus, the numerical result, [tex]\( (-0.5, 0.5) \)[/tex], implies:
- The wrong result by Henry is [tex]\( \frac{1}{2} \)[/tex] ([tex]\( 0.5 \)[/tex] in decimal).
- The correct result is [tex]\( -\frac{1}{2} \)[/tex] ([tex]\(-0.5\)[/tex] in decimal).
### Henry's Steps:
1. Subtract [tex]\( 2 \pi \)[/tex] from [tex]\( \frac{10 \pi}{3} \)[/tex] as many times as possible:
[tex]\[ \frac{10 \pi}{3} - 2 \pi = \frac{10 \pi}{3} - \frac{6 \pi}{3} = \frac{4 \pi}{3} \][/tex]
This step is correct. [tex]\( \frac{10 \pi}{3} \)[/tex] reduced by cycle of [tex]\( 2 \pi \)[/tex] lands at [tex]\( \frac{4 \pi}{3} \)[/tex].
2. Find the reference angle for [tex]\( \frac{4 \pi}{3} \)[/tex]:
Here, Henry states:
[tex]\[ \frac{3 \pi}{2} - \frac{4 \pi}{3} = \frac{\pi}{6} \][/tex]
However, this approach is incorrect. The correct way to find the reference angle for [tex]\( \frac{4 \pi}{3} \)[/tex] is to subtract [tex]\( \pi \)[/tex]:
[tex]\[ \text{Reference angle} = \frac{4 \pi}{3} - \pi = \frac{4 \pi}{3} - \frac{3 \pi}{3} = \frac{\pi}{3} \][/tex]
3. The cosine value for [tex]\( \frac{\pi}{6} \)[/tex] is [tex]\( \frac{\sqrt{3}}{2} \)[/tex]:
While this cosine value is correct for [tex]\( \frac{\pi}{6} \)[/tex], the reference angle [tex]\( \frac{\pi}{6} \)[/tex] is wrong.
4. The cosine value is positive because [tex]\( \frac{\pi}{6} \)[/tex] is in the first quadrant:
This conclusion is also incorrect because [tex]\( \frac{4 \pi}{3} \)[/tex] is located in the third quadrant where cosine values are negative.
### Correct Procedure:
Let's correct the solution step-by-step:
1. Reduce the angle:
[tex]\[ \frac{10 \pi}{3} \equiv \frac{4 \pi}{3} \, (\text{mod} \, 2 \pi) \][/tex]
This reduction is correct.
2. Determine the quadrant:
[tex]\( \frac{4 \pi}{3} \)[/tex] is in the third quadrant of the unit circle.
3. Find the reference angle:
[tex]\[ \text{Reference angle} = \frac{4 \pi}{3} - \pi = \frac{\pi}{3} \][/tex]
4. Evaluate the cosine for the reference angle in the third quadrant:
[tex]\[ \cos \frac{\pi}{3} = \frac{1}{2} \][/tex]
Since cosine is negative in the third quadrant:
[tex]\[ \cos \frac{4 \pi}{3} = -\frac{1}{2} \][/tex]
Therefore, the correct value of [tex]\( \cos \frac{4 \pi}{3} \)[/tex] is [tex]\( -\frac{1}{2} \)[/tex].
### Identifying Henry's Errors:
- Henry incorrectly calculated the reference angle. Instead of using [tex]\( \frac{4 \pi}{3} - \pi \)[/tex], he incorrectly used [tex]\( \frac{3 \pi}{2} \)[/tex].
- He misunderstood the quadrant properties for cosine. In the third quadrant, cosine values should be negative.
### Correct Answer:
- Henry’s reference angle calculation is incorrect.
- Henry’s determination of the cosine sign is incorrect.
The correct value of [tex]\( \cos \frac{10 \pi}{3} \)[/tex] is [tex]\( -\frac{1}{2} \)[/tex].
Thus, the numerical result, [tex]\( (-0.5, 0.5) \)[/tex], implies:
- The wrong result by Henry is [tex]\( \frac{1}{2} \)[/tex] ([tex]\( 0.5 \)[/tex] in decimal).
- The correct result is [tex]\( -\frac{1}{2} \)[/tex] ([tex]\(-0.5\)[/tex] in decimal).
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
