Question

If cotθ=3/4 and the terminal point determined by θ is in quadrant 3, then: (choose all that apply)

Answer 1

With $\theta$ in quadrant 3, we should expect both $\cos\theta$ and $\sin\theta$ to be negative, so that $\tan\theta$ is positive. The corresponding reciprocal expressions $(\sec\theta,\csc\theta,\cot\theta)$ will have the same sign.

$\cot\theta=\dfrac34\implies\tan\theta=\dfrac43$

Recall that $1+\tan^2\theta=\sec^2\theta$, which means

$\sec\theta=-\sqrt{1+\tan^2\theta}=-\dfrac53$

$\implies\cos\theta=-\dfrac35$

Also recall that $\cos^2\theta+\sin^2\theta=1$, so

$\sin\theta=-\sqrt{1-\cos^2\theta}=-\dfrac45$

$\implies\csc\theta=-\dfrac54$

Only the first two options are correct.