Question

Let θ be an angle in Quadrant I such that θ=tan^-1(4/3). What is the value of csc θ?

Answer 1

If $\theta=\tan^{-1}\dfrac43$, then $\tan\theta=\dfrac43$ and so $\cot\theta=\dfrac34$. Recall the Pythagorean identity,

$\csc^2\theta=\cot^2\theta+1\implies\csc\theta=\pm\sqrt{\cot^2\theta+1}$

$\theta$ lies in the first quadrant, so we know $\sin\theta>0$, which also means $\csc\theta=\dfrac1{\sin\theta}>0$, so we should take the positive square root. Then

$\csc\theta=\sqrt{\left(\dfrac34\right)^2+1}=\dfrac54$