Sure, let's analyze the given trigonometric values step by step.
1. [tex]\(\sin\left(\frac{\pi}{2}\right)\)[/tex]:
The sine of [tex]\(\frac{\pi}{2}\)[/tex] (90 degrees) is 1. This is because on the unit circle, the coordinates at [tex]\(\frac{\pi}{2}\)[/tex] are (0, 1), and sine corresponds to the y-coordinate. Thus:
[tex]\[
\sin\left(\frac{\pi}{2}\right) = 1
\][/tex]
2. [tex]\(\tan(\pi)\)[/tex]:
The tangent of [tex]\(\pi\)[/tex] (180 degrees) is 0. For tangent, we use the ratio of sine over cosine:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
At [tex]\(\pi\)[/tex], the coordinates are (-1, 0) on the unit circle. So:
[tex]\[
\tan(\pi) = \frac{\sin(\pi)}{\cos(\pi)} = \frac{0}{-1} = 0
\][/tex]
3. [tex]\(\tan\left(\frac{3\pi}{2}\right)\)[/tex]:
The tangent of [tex]\(\frac{3\pi}{2}\)[/tex] (270 degrees) is undefined, which we generally consider as [tex]\(\infty\)[/tex] (infinity). On the unit circle, the coordinates are (0, -1), and when we calculate:
[tex]\[
\tan\left(\frac{3\pi}{2}\right) = \frac{\sin\left(\frac{3\pi}{2}\right)}{\cos\left(\frac{3\pi}{2}\right)} = \frac{-1}{0}
\][/tex]
Division by zero is undefined, hence we consider it as [tex]\(\infty\)[/tex].
4. [tex]\(\cos(2\pi)\)[/tex]:
The cosine of [tex]\(2\pi\)[/tex] (360 degrees) is 1. For cosine, we take the x-coordinate of the point on the unit circle. At [tex]\(2\pi\)[/tex], the coordinates are (1, 0). So:
[tex]\[
\cos(2\pi) = 1
\][/tex]
In summary:
[tex]\[
\begin{array}{l}
\sin\left(\frac{\pi}{2}\right) = 1 \\
\tan(\pi) = 0 \\
\tan\left(\frac{3\pi}{2}\right) = \infty \\
\cos(2\pi) = 1
\end{array}
\][/tex]
