## Trigonometry (11th Edition) Clone

$$\cos\theta=\frac{24}{25}$$ $$\tan\theta=-\frac{7}{24}$$ $$\cot\theta=-\frac{24}{7}$$ $$\sec\theta=\frac{25}{24}$$ $$\csc\theta=-\frac{25}{7}$$
$$\sin\theta=-\frac{7}{25}\hspace{1.5cm}\theta\hspace{.1cm}\text{in quadrant IV}$$ $\theta$ is in quadrant IV, meaning that $\cos\theta\gt0$. 1) First, apply Pythagorean Identity to find $\cos\theta$ $$\cos^2\theta=1-\sin^2\theta=1-\Big(-\frac{7}{25}\Big)^2$$ $$\cos^2\theta=1-\Big(\frac{49}{625}\Big)=\frac{576}{625}$$ So, $$\cos\theta=\pm\frac{24}{25}$$ But since $\cos\theta\gt0$, $$\cos\theta=\frac{24}{25}$$ 2) Apply Quotient Identities to find $\tan\theta$ and $\cot\theta$ $$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{-\frac{7}{25}}{\frac{24}{25}}=-\frac{7}{24}$$ $$\cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{\frac{24}{25}}{-\frac{7}{25}}=-\frac{24}{7}$$ 3) Apply Reciprocal Identities to find $\sec\theta$ and $\csc\theta$ $$\sec\theta=\frac{1}{\cos\theta}=\frac{1}{\frac{24}{25}}=\frac{25}{24}$$ $$\csc\theta=\frac{1}{\sin\theta}=\frac{1}{-\frac{7}{25}}=-\frac{25}{7}$$