Answer
$$\sin\theta=-\frac{\sqrt7}{4}$$
$$\cos\theta=\frac{3}{4}$$
$$\csc\theta=-\frac{4\sqrt7}{7}$$
$$\tan\theta=-\frac{\sqrt7}{3}$$
$$\cot\theta=-\frac{3\sqrt7}{7}$$
Work Step by Step
$$\sec\theta=\frac{4}{3}\hspace{1.5cm}\sin\theta\lt0$$
1) Find $\cos\theta$
- Reciprocal Identities:
$$\sec\theta=\frac{1}{\cos\theta}$$
$$\cos\theta=\frac{1}{\sec\theta}=\frac{1}{\frac{4}{3}}=\frac{3}{4}$$
2) Find $\sin\theta$
- Pythagorean Identities:
$$\sin^2\theta=1-\cos^2\theta=1-(\frac{3}{4})^2=1-\frac{9}{16}=\frac{7}{16}$$
$$\sin\theta=\pm\frac{\sqrt7}{4}$$
But since $\sin\theta\lt0$,
$$\sin\theta=-\frac{\sqrt7}{4}$$
3) Find $\csc\theta$
- Reciprocal Identities:
$$\csc\theta=\frac{1}{\sin\theta}=\frac{1}{-\frac{\sqrt7}{4}}=-\frac{4}{\sqrt7}=-\frac{4\sqrt7}{7}$$
4) Find $\tan\theta$ and $\cot\theta$
- Quotient Identities:
$$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{-\frac{\sqrt7}{4}}{\frac{3}{4}}=-\frac{\sqrt7}{3}$$
- Reciprocal Identities:
$$\cot\theta=\frac{1}{\tan\theta}=\frac{1}{-\frac{\sqrt7}{3}}=-\frac{3}{\sqrt7}=-\frac{3\sqrt7}{7}$$