Answer
$\frac{\sqrt 3}{5}, -\frac{\sqrt {22}}{5}, -\frac{\sqrt {66}}{22}, -\frac{\sqrt {66}}{3}, -\frac{\sqrt {110}}{22}, \frac{5\sqrt {3}}{3}$
Work Step by Step
1. Given $sin\theta=\frac{\sqrt 3}{5}$ and $cos\theta\lt0$, we can find that $\theta$ is in quadrant II. Form a right triangle with sides $\sqrt 3, \sqrt {5^2-3}, 5$ or $\sqrt 3,\sqrt {22},5$, we have:
2. $sin\theta=\frac{\sqrt 3}{5}$
3. $cos\theta=-\frac{\sqrt {22}}{5}$
4. $tan\theta=-\frac{\sqrt {3}}{\sqrt {22}}=-\frac{\sqrt {66}}{22}$
5. $cot\theta=\frac{1}{tan\theta}=-\frac{\sqrt {66}}{3}$
6. $sec\theta=\frac{1}{cos\theta}=-\frac{\sqrt {110}}{22}$
7. $csc\theta=\frac{1}{sin\theta}=\frac{5\sqrt {3}}{3}$
