find tan and cot if cos = -5/9 and cot <0

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.

This solution explains how to find the values of tan(θ) and cot(θ) given cos(θ) = -5/9 and cot(θ) < 0, using trigonometric identities and the Pythagorean theorem. Suitable for high school students in Grades 10-12.

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