Given that $\cos \theta = \frac{-5}{9}$ and $\cot \theta < 0$, we can find both $\tan \theta$ and $\cot \theta$.

Step 1: Determine the quadrant

• Since $\cos \theta = \frac{-5}{9}$, the cosine is negative.
• $\cot \theta < 0$ implies that $\tan \theta$ is also negative (since $\cot \theta = \frac{1}{\tan \theta}$).

Cosine is negative and tangent is negative in the second quadrant.

Step 2: Find $\sin \theta$ using the Pythagorean identity

The identity $\sin^2 \theta + \cos^2 \theta = 1$ gives us:

$\sin^2 \theta + \left( \frac{-5}{9} \right)^2 = 1$ $\sin^2 \theta + \frac{25}{81} = 1$ $\sin^2 \theta = 1 - \frac{25}{81}$ $\sin^2 \theta = \frac{81}{81} - \frac{25}{81} = \frac{56}{81}$ $\sin \theta = \pm \sqrt{\frac{56}{81}} = \pm \frac{\sqrt{56}}{9}$

Since we are in the second quadrant, where sine is positive, we have: $\sin \theta = \frac{\sqrt{56}}{9}$ Simplifying $\sqrt{56}$: $\sin \theta = \frac{2\sqrt{14}}{9}$

Step 3: Find $\tan \theta$

Now, we can use the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$:

$\tan \theta = \frac{\frac{2\sqrt{14}}{9}}{\frac{-5}{9}} = \frac{2\sqrt{14}}{-5} = -\frac{2\sqrt{14}}{5}$

Step 4: Find $\cot \theta$

Since $\cot \theta = \frac{1}{\tan \theta}$, we have: $\cot \theta = \frac{1}{-\frac{2\sqrt{14}}{5}} = -\frac{5}{2\sqrt{14}}$

You can rationalize the denominator if needed: $\cot \theta = -\frac{5}{2\sqrt{14}} \times \frac{\sqrt{14}}{\sqrt{14}} = -\frac{5\sqrt{14}}{28}$

Final Answer:

• $\tan \theta = -\frac{2\sqrt{14}}{5}$
• $\cot \theta = -\frac{5\sqrt{14}}{28}$

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

Relative Questions:

1. How can you use the Pythagorean identity to find $\sin \theta$ given $\cos \theta$?
2. In which quadrants are both sine and tangent negative?
3. How does the sign of cotangent affect the quadrant of $\theta$?
4. What is the process of rationalizing the denominator, and why is it done?
5. Can you derive the Pythagorean identities from the unit circle?

Tip:

When determining the quadrant of an angle, remember that cosine, sine, and tangent each have different signs in different quadrants—this helps you identify which trigonometric functions are positive or negative.