No multiple choice, just an answer box... Help?

Answer:[tex]csc(\theta)=\frac{5}{3}[/tex]Step-by-step explanation:If the [tex]tan(\theta)=\frac{3}{4}[/tex], then the cotangent is [tex]\frac{4}{3}[/tex], given that:[tex]tan(\theta)=\frac{sin(\theta)}{cos(\theta)}[/tex], and [tex]cot(\theta)=\frac{cos(\theta)}{sin(\theta)}[/tex].This is important to know because if you recall the three Pythagorean identities in trigonometry, one of them involves a nice relationship between the cotangent and the cosecant of an angle:[tex]1+cot^2(\theta)=csc^2(\theta)[/tex]so we can replace [tex]cot(\theta)[/tex] with 4/3, and find what [tex]csc(\theta)[/tex] is using that identity:[tex]1+cot^2(\theta)=csc^2(\theta)\\1+(\frac{4}{3})^2= csc^2(\theta)\\1+\frac{16}{9} =csc^2(\theta)\\\frac{25}{9} = csc^2(\theta)\\csc(\theta)=+/-\sqrt{\frac{25}{9}} \\csc(\theta)=+/-\frac{5}{3}[/tex]Now, you have to decide on which sign to use. So consider that if the tangent was positive, so most likely you are dealing with an angle [tex]\theta[/tex] between 0 and [tex]\frac{\pi }{2}[/tex], and in that quadrant, the cosecant is positive.Therefore, pick the positive value: 5/3