Question

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

Answer 1

Answer:

$csc(\theta)=\frac{5}{3}$

Step-by-step explanation:

If the $tan(\theta)=\frac{3}{4}$, then the cotangent is $\frac{4}{3}$, given that:

$tan(\theta)=\frac{sin(\theta)}{cos(\theta)}$, and $cot(\theta)=\frac{cos(\theta)}{sin(\theta)}$.

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:

$1+cot^2(\theta)=csc^2(\theta)$

so we can replace $cot(\theta)$ with 4/3, and find what $csc(\theta)$ is using that identity:

$1+cot^2(\theta)=csc^2(\theta)\\1+(\frac{4}{3})^2= csc^2(\theta)\\1+\frac{16}{9} =csc^2(\theta)\\\frac{25}{9} = csc^2(\theta)\\csc(\theta)=+/-\sqrt{\frac{25}{9}} \\csc(\theta)=+/-\frac{5}{3}$

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 $\theta$ between 0 and $\frac{\pi }{2}$, and in that quadrant, the cosecant is positive.

Therefore, pick the positive value: 5/3