Answer
$$\sin\theta=\frac{3}{5}$$
$$\cos\theta=\frac{4}{5}$$
$$\tan\theta=\frac{3}{4}$$
$$\sec\theta=\frac{5}{4}$$
$$\csc\theta=\frac{5}{3}$$
Work Step by Step
$$\cot\theta=\frac{4}{3}\hspace{1.5cm}\sin\theta\gt0$$
1) Reciprocal Identities:
$$\cot\theta=\frac{1}{\tan\theta}$$
$$\tan\theta=\frac{1}{\cot\theta}=\frac{1}{\frac{4}{3}}=\frac{3}{4}$$
2) Pythagorean Identities:
$$\csc^2\theta=\cot^2\theta+1=(\frac{4}{3})^2+1=\frac{16}{9}+1=\frac{25}{9}$$
$$\csc\theta=\pm\frac{5}{3}$$
$$\sec^2\theta=\tan^2\theta+1=(\frac{3}{4})^2+1=\frac{9}{16}+1=\frac{25}{16}$$
$$\sec\theta=\pm\frac{5}{4}$$
3) Reciprocal Identities:
$$\csc\theta=\frac{1}{\sin\theta}$$
As $\sin\theta\gt0$, this means that $\csc\theta\gt0$
$$\csc\theta=\frac{5}{3}$$
Also, using a Quotient Identity, we have $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$
As $\tan\theta=\frac{3}{4}\gt0$ and $\sin\theta\gt0$, $\cos\theta\gt0$.
Furthermore, $$\sec\theta=\frac{1}{\cos\theta}$$
Since $\cos\theta\gt0$, this also means that $\sec\theta\gt0$.
$$\sec\theta=\frac{5}{4}$$
Now for $\sin\theta$ and $\cos\theta$:
$$\sin\theta=\frac{1}{\csc\theta}=\frac{1}{\frac{5}{3}}=\frac{3}{5}$$
$$\cos\theta=\frac{1}{\sec\theta}=\frac{1}{\frac{5}{4}}=\frac{4}{5}$$
