To determine which trigonometric functions have a value of 0, we need to evaluate each one from the list provided:
1. [tex]\(\cos \frac{\pi}{2}\)[/tex]
2. [tex]\(\cos 0\)[/tex]
3. [tex]\(\sin 0\)[/tex]
4. [tex]\(\sin \frac{3\pi}{2}\)[/tex]
5. [tex]\(\tan \pi\)[/tex]
Let's evaluate these trigonometric functions:
1. [tex]\(\cos \frac{\pi}{2}\)[/tex]:
[tex]\[
\cos \frac{\pi}{2} \approx 6.123233995736766 \times 10^{-17}
\][/tex]
This value is extremely close to 0, and for practical purposes, it can be considered as 0 due to the inherent computational precision.
2. [tex]\(\cos 0\)[/tex]:
[tex]\[
\cos 0 = 1.0
\][/tex]
This value is not 0.
3. [tex]\(\sin 0\)[/tex]:
[tex]\[
\sin 0 = 0.0
\][/tex]
This value is 0.
4. [tex]\(\sin \frac{3\pi}{2}\)[/tex]:
[tex]\[
\sin \frac{3\pi}{2} = -1.0
\][/tex]
This value is not 0.
5. [tex]\(\tan \pi\)[/tex]:
[tex]\[
\tan \pi \approx -1.2246467991473532 \times 10^{-16}
\][/tex]
Similar to [tex]\(\cos \frac{\pi}{2}\)[/tex], this value is extremely close to 0, and it can be considered as 0 due to the precision limitations.
Therefore, the trigonometric functions that have a value of 0 are:
1. [tex]\(\cos \frac{\pi}{2}\)[/tex]
2. [tex]\(\sin 0\)[/tex]
3. [tex]\(\tan \pi\)[/tex]
So, the correct answers are:
- [tex]\(\cos \frac{\pi}{2}\)[/tex]
- [tex]\(\sin 0\)[/tex]
- [tex]\(\tan \pi\)[/tex]
