Answer:
![\csc(x)=5/3](https://tex.z-dn.net/?f=%5Ccsc%28x%29%3D5%2F3)
Step-by-step explanation:
So we know that:
![\tan(x)=3/4](https://tex.z-dn.net/?f=%5Ctan%28x%29%3D3%2F4)
Recall that tangent represents the side opposite over the side adjacent.
This means that the opposite side is 3 while the adjacent side is 4.
Therefore, by using the Pythagorean Theorem, we can solve for the third side, which is the hypotenuse:
![a^2+b^2=c^2](https://tex.z-dn.net/?f=a%5E2%2Bb%5E2%3Dc%5E2)
Substitute a for 3 and b for 4. Thus:
![3^2+4^2=c^2](https://tex.z-dn.net/?f=3%5E2%2B4%5E2%3Dc%5E2)
Square and add:
![9+16=c^2\\25=c^2](https://tex.z-dn.net/?f=9%2B16%3Dc%5E2%5C%5C25%3Dc%5E2)
Square root:
![c=5](https://tex.z-dn.net/?f=c%3D5)
Now, recall that cosecant is the reciprocal of sine. So, find sine first.
Sine is opposite over hypotenuse. From tangent, the opposite is 3 and the hypotenuse as we now know is 5. Thus:
![\sin(x)=3/5](https://tex.z-dn.net/?f=%5Csin%28x%29%3D3%2F5)
And cosecant is the reciprocal of that. Thus:
![\csc(x)=5/3](https://tex.z-dn.net/?f=%5Ccsc%28x%29%3D5%2F3)
And that's our answer :)