Answer :
We need to find the corresponding point on the unit circle for the given radian measure [tex]\(\theta = \frac{7\pi}{6}\)[/tex].
Let’s go through the steps:
1. Evaluate the angle:
[tex]\(\theta = \frac{7\pi}{6}\)[/tex] is in the third quadrant because [tex]\(\frac{7\pi}{6}\)[/tex] radians is just slightly more than [tex]\(\pi\)[/tex] (which is [tex]\(\frac{6\pi}{6}\)[/tex]).
2. Determine the cosine and sine values:
To find the coordinates of the corresponding point on the unit circle, we use the cosine and sine functions. For an angle [tex]\(\theta\)[/tex]:
[tex]\[ x = \cos(\theta) \][/tex]
[tex]\[ y = \sin(\theta) \][/tex]
3. Identify the exact values based on the unit circle:
Since [tex]\(\theta = \frac{7\pi}{6}\)[/tex] is in the third quadrant, both sine and cosine values will be negative. For an angle of [tex]\(\frac{\pi}{6}\)[/tex], the reference angle of [tex]\(\frac{7\pi}{6} - \pi = \frac{\pi}{6}\)[/tex], the values of cosine and sine are:
[tex]\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \][/tex]
[tex]\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \][/tex]
In the third quadrant, these values become negative:
[tex]\[ \cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2} \][/tex]
[tex]\[ \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2} \][/tex]
4. Identify the corresponding point:
Therefore, the point on the unit circle corresponding to [tex]\(\theta = \frac{7\pi}{6}\)[/tex] is:
[tex]\[ \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \][/tex]
5. Match it with the given choices:
A. [tex]\(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex]
B. [tex]\(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)[/tex]
C. [tex]\(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)[/tex]
D. [tex]\(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)[/tex]
The correct answer is:
D. [tex]\(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)[/tex]
Let’s go through the steps:
1. Evaluate the angle:
[tex]\(\theta = \frac{7\pi}{6}\)[/tex] is in the third quadrant because [tex]\(\frac{7\pi}{6}\)[/tex] radians is just slightly more than [tex]\(\pi\)[/tex] (which is [tex]\(\frac{6\pi}{6}\)[/tex]).
2. Determine the cosine and sine values:
To find the coordinates of the corresponding point on the unit circle, we use the cosine and sine functions. For an angle [tex]\(\theta\)[/tex]:
[tex]\[ x = \cos(\theta) \][/tex]
[tex]\[ y = \sin(\theta) \][/tex]
3. Identify the exact values based on the unit circle:
Since [tex]\(\theta = \frac{7\pi}{6}\)[/tex] is in the third quadrant, both sine and cosine values will be negative. For an angle of [tex]\(\frac{\pi}{6}\)[/tex], the reference angle of [tex]\(\frac{7\pi}{6} - \pi = \frac{\pi}{6}\)[/tex], the values of cosine and sine are:
[tex]\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \][/tex]
[tex]\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \][/tex]
In the third quadrant, these values become negative:
[tex]\[ \cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2} \][/tex]
[tex]\[ \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2} \][/tex]
4. Identify the corresponding point:
Therefore, the point on the unit circle corresponding to [tex]\(\theta = \frac{7\pi}{6}\)[/tex] is:
[tex]\[ \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \][/tex]
5. Match it with the given choices:
A. [tex]\(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex]
B. [tex]\(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)[/tex]
C. [tex]\(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)[/tex]
D. [tex]\(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)[/tex]
The correct answer is:
D. [tex]\(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)[/tex]
