Answer:
[tex]\csc(x)=5/3[/tex]
Step-by-step explanation:
So we know that:
[tex]\tan(x)=3/4[/tex]
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:
[tex]a^2+b^2=c^2[/tex]
Substitute a for 3 and b for 4. Thus:
[tex]3^2+4^2=c^2[/tex]
Square and add:
[tex]9+16=c^2\\25=c^2[/tex]
Square root:
[tex]c=5[/tex]
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:
[tex]\sin(x)=3/5[/tex]
And cosecant is the reciprocal of that. Thus:
[tex]\csc(x)=5/3[/tex]
And that's our answer :)
