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The given problem provides two conditions: 

1. \( \tan \theta = \frac{\sqrt{11}}{5} \)
2. \( \cos \theta < 0 \)

You are tasked with finding the exact values of the five trigonometric ratios not given (i.e., \(\sin \theta\), \(\cos \theta\), \(\sec \theta\), \(\csc \theta\), and \(\cot \theta\)).

### Step-by-Step Solution:

#### Step 1: Use the identity relating \(\tan \theta\)
We know:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]
Given \( \tan \theta = \frac{\sqrt{11}}{5} \), this implies:
\[
\frac{\sin \theta}{\cos \theta} = \frac{\sqrt{11}}{5}
\]
Thus, \( \sin \theta = \frac{\sqrt{11}}{5} \cos \theta \).

#### Step 2: Use the Pythagorean identity
We can use the identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Substitute \( \sin \theta = \frac{\sqrt{11}}{5} \cos \theta \) into this equation:
\[
\left( \frac{\sqrt{11}}{5} \cos \theta \right)^2 + \cos^2 \theta = 1
\]
\[
\frac{11}{25} \cos^2 \theta + \cos^2 \theta = 1
\]
Factor out \( \cos^2 \theta \):
\[
\cos^2 \theta \left( \frac{11}{25} + 1 \right) = 1
\]
\[
\cos^2 \theta \left( \frac{11}{25} + \frac{25}{25} \right) = 1
\]
\[
\cos^2 \theta \times \frac{36}{25} = 1
\]
\[
\cos^2 \theta = \frac{25}{36}
\]
Thus:
\[
\cos \theta = -\frac{5}{6}
\]
(since it is given that \( \cos \theta < 0 \)).

#### Step 3: Find \(\sin \theta\)
Now that we know \( \cos \theta = -\frac{5}{6} \), we can substitute this into the equation for \( \sin \theta \):
\[
\sin \theta = \frac{\sqrt{11}}{5} \times \left( -\frac{5}{6} \right) = -\frac{\sqrt{11}}{6}
\]

#### Step 4: Find the remaining trigonometric ratios
1. **\(\sec \theta\)**:
   \[
   \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{5}{6}} = -\frac{6}{5}
   \]
2. **\(\csc \theta\)**:
   \[
   \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{\sqrt{11}}{6}} = -\frac{6}{\sqrt{11}} = -\frac{6\sqrt{11}}{11}
   \]
3. **\(\cot \theta\)**:
   \[
   \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{\sqrt{11}}{5}} = \frac{5}{\sqrt{11}} = \frac{5\sqrt{11}}{11}
   \]

### Final Answer:
- \( \sin \theta = -\frac{\sqrt{11}}{6} \)
- \( \cos \theta = -\frac{5}{6} \)
- \( \sec \theta = -\frac{6}{5} \)
- \( \csc \theta = -\frac{6\sqrt{11}}{11} \)
- \( \cot \theta = \frac{5\sqrt{11}}{11} \)

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. How does the unit circle relate to the signs of trigonometric functions in different quadrants?
2. How would the values change if \( \tan \theta \) was negative instead of positive?
3. What are the identities involving the sum and difference of angles for \(\sin\) and \(\cos\)?
4. How can the double-angle identity for tangent help in problems like this?
5. What is the importance of knowing the domain restrictions for trigonometric functions?

**Tip:** Always check the quadrant to determine the signs of trigonometric values when dealing with problems like this.

Find the exact values of the five trigonometric ratios not given: tan(θ) = √11/5 and cos(θ) < 0

Learn how to find the exact values of sine, cosine, secant, cosecant, and cotangent when tan(θ) = √11/5 and cos(θ) < 0. This problem involves key trigonometric identities and the Pythagorean Theorem. Suitable for high school students in Grades 10-12.

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