We are given that $\sin \theta = -\frac{3}{5}$ and that $\theta$ is in quadrant III, where both sine and cosine are negative. We need to find the exact values of $\sec \theta$ and $\cot \theta$.

Step 1: Find $\cos \theta$

Using the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$ Substitute $\sin \theta = -\frac{3}{5}$: $\left( -\frac{3}{5} \right)^2 + \cos^2 \theta = 1$ $\frac{9}{25} + \cos^2 \theta = 1$ $\cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}$ $\cos \theta = -\frac{4}{5}$ (We take the negative root because $\theta$ is in quadrant III, where cosine is negative.)

Step 2: Find $\sec \theta$

$\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}$

Step 3: Find $\cot \theta$

Using the identity $\cot \theta = \frac{\cos \theta}{\sin \theta}$: $\cot \theta = \frac{-\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{3}$

Final Answers:

• $\sec \theta = -\frac{5}{4}$
• $\cot \theta = \frac{4}{3}$

Would you like more details on any step, or do you have questions?

Here are 5 related questions:

1. How do you determine the signs of trigonometric functions in different quadrants?
2. What is the significance of using the Pythagorean identity in trigonometry?
3. Can you explain how to derive $\sec \theta$ from $\cos \theta$?
4. How does the quadrant affect the value of $\cot \theta$?
5. What other identities can be used to calculate secant and cotangent?

Tip: Remember that in quadrant III, both sine and cosine are negative, while tangent and cotangent are positive.