1. **State the problem:** We need to find the angles on the unit circle that correspond to a given reciprocal trigonometric ratio (cosecant, secant, or cotangent). 2. **Recall the reciprocal trig functions:** - Cosecant: $\csc \theta = \frac{1}{\sin \theta}$ - Secant: $\sec \theta = \frac{1}{\cos \theta}$ - Cotangent: $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$ 3. **Important rules:** - The unit circle has angles from $0$ to $2\pi$ radians (or $0^\circ$ to $360^\circ$). - Reciprocal functions are undefined where their base functions are zero (e.g., $\csc \theta$ undefined when $\sin \theta = 0$). 4. **Find angles for a given reciprocal ratio:** - Suppose the given reciprocal ratio is $r$ for $\csc \theta$. - Then $\sin \theta = \frac{1}{r}$. - Find $\theta$ such that $\sin \theta = \frac{1}{r}$. - Use inverse sine: $\theta = \sin^{-1}\left(\frac{1}{r}\right)$. 5. **Find all solutions on the unit circle:** - Since $\sin \theta$ is positive in Quadrants I and II, the solutions are: $$\theta_1 = \sin^{-1}\left(\frac{1}{r}\right), \quad \theta_2 = \pi - \sin^{-1}\left(\frac{1}{r}\right)$$ - Similarly for $\sec \theta = r$, $\cos \theta = \frac{1}{r}$, solutions are: $$\theta_1 = \cos^{-1}\left(\frac{1}{r}\right), \quad \theta_2 = 2\pi - \cos^{-1}\left(\frac{1}{r}\right)$$ - For $\cot \theta = r$, $\tan \theta = \frac{1}{r}$, solutions are: $$\theta_1 = \tan^{-1}\left(\frac{1}{r}\right), \quad \theta_2 = \pi + \tan^{-1}\left(\frac{1}{r}\right)$$ 6. **Summary:** To find angles with a given reciprocal trig ratio $r$, invert the reciprocal to get the base trig ratio, then find all angles on the unit circle where the base trig function equals that value. This method applies to any reciprocal trig ratio given.