## Trigonometry (11th Edition) Clone

$$\sin\theta=\frac{3}{5}$$ $$\cos\theta=\frac{4}{5}$$ $$\tan\theta=\frac{3}{4}$$ $$\sec\theta=\frac{5}{4}$$ $$\csc\theta=\frac{5}{3}$$
$$\cot\theta=\frac{4}{3}\hspace{1.5cm}\sin\theta\gt0$$ 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 $\cos\theta$: $$\sin\theta=\frac{1}{\csc\theta}=\frac{1}{\frac{5}{3}}=\frac{3}{5}$$ $$\cos\theta=\frac{1}{\sec\theta}=\frac{1}{\frac{5}{4}}=\frac{4}{5}$$