Select all that have a value of 0.

A. [tex]\cos \frac{\pi}{2}[/tex]

B. [tex]\cos 0[/tex]

C. [tex]\sin 0[/tex]

D. [tex]\sin \frac{3 \pi}{2}[/tex]

E. [tex]\tan \pi[/tex]



Answer :

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]