Let's solve for [tex]\(\sin(\theta)\)[/tex] and [tex]\(\tan(\theta)\)[/tex] given that [tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex] and [tex]\(\frac{3\pi}{2} < \theta < 2\pi\)[/tex].
### Step-by-Step Solution:
1. Identify the Quadrant:
Given that [tex]\(\frac{3\pi}{2} < \theta < 2\pi\)[/tex], we know that [tex]\(\theta\)[/tex] is in the fourth quadrant.
2. Determine [tex]\(\theta\)[/tex]:
In the fourth quadrant, [tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex] corresponds to an angle of [tex]\(\theta = \frac{7\pi}{4}\)[/tex]. This angle satisfies the given range because [tex]\( \frac{7\pi}{4} \)[/tex] (which is [tex]\(315^\circ\)[/tex]) is between [tex]\( 270^\circ (\frac{3\pi}{2}) \)[/tex] and [tex]\( 360^\circ (2\pi) \)[/tex].
3. Calculate [tex]\(\sin(\theta)\)[/tex]:
In the fourth quadrant, the sine function is negative. The value [tex]\(\theta = \frac{7\pi}{4}\)[/tex] corresponds to an angle where sine value is:
[tex]\[
\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\][/tex]
4. Calculate [tex]\(\tan(\theta)\)[/tex]:
The tangent function is the ratio of sine and cosine:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
Given [tex]\(\sin(\theta) = -\frac{\sqrt{2}}{2}\)[/tex] and [tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[
\tan\left(\frac{7\pi}{4}\right) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1
\][/tex]
### Conclusion:
- [tex]\(\sin(\theta) = -\frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\tan(\theta) = -1\)[/tex]
Thus, the numerical values are:
- [tex]\(\sin(\theta) = -0.7071067811865477\)[/tex]
- [tex]\(\tan(\theta) = -1.0000000000000004\)[/tex]
Answer:
For [tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex] and [tex]\(\frac{3\pi}{2} < \theta < 2\pi\)[/tex]:
[tex]\(\sin(\theta) = -\frac{\sqrt{2}}{2}\)[/tex]
[tex]\(\tan(\theta) = -1\)[/tex]
