Question

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

Answer 1

`csc(\theta)=(5)/(3)`

Explanation:

If the
`tan(\theta)=(3)/(4)`, then the cotangent is
`(4)/(3)`, given that:

`tan(\theta)=(sin(\theta))/(cos(\theta))`, and
`cot(\theta)=(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+((4)/(3))^2= csc^2(\theta)\\1+(16)/(9) =csc^2(\theta)\\(25)/(9) = csc^2(\theta)\\csc(\theta)=+/-\sqrt{(25)/(9)} \\csc(\theta)=+/-(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
`(\pi )/(2)`, and in that quadrant, the cosecant is positive.

Therefore, pick the positive value: 5/3