Answer
$$\cos\theta=-\frac{3}{5}$$
$$\tan\theta=\frac{4}{3}$$
$$\cot\theta=\frac{3}{4}$$
$$\sec\theta=-\frac{5}{3}$$
$$\csc\theta=-\frac{5}{4}$$
Work Step by Step
$$\sin\theta=-\frac{4}{5}\hspace{1.5cm}\cos\theta\lt0$$
1) Find $\cos\theta$
- Pythagorean Identities:
$$\cos^2\theta=1-\sin^2\theta=1-(-\frac{4}{5})^2=1-\frac{16}{25}=\frac{9}{25}$$
$$\cos\theta=\pm\frac{3}{5}$$
But since $\cos\theta\lt0$, $$\cos\theta=-\frac{3}{5}$$
2) Find $\tan\theta$ and $\cot\theta$
- Quotient Identities:
$$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{-\frac{4}{5}}{-\frac{3}{5}}=\frac{4}{3}$$
$$\cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{-\frac{3}{5}}{-\frac{4}{5}}=\frac{3}{4}$$
3) Find $\sec\theta$ and $\csc\theta$
- Reciprocal Identities:
$$\sec\theta=\frac{1}{\cos\theta}=\frac{1}{-\frac{3}{5}}=-\frac{5}{3}$$
$$\csc\theta=\frac{1}{\sin\theta}=\frac{1}{-\frac{4}{5}}=-\frac{5}{4}$$
