knowing sin=-(\sqrt(7))/(4), calculate cota

Given that \(\sin(\theta) = -\frac{\sqrt{7}}{4}\), we want to calculate \(\cot(\theta)\).

### Step 1: Use the Pythagorean identity to find \(\cos(\theta)\).
The identity is:

\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]

Substitute \(\sin(\theta) = -\frac{\sqrt{7}}{4}\):

\[
\left(-\frac{\sqrt{7}}{4}\right)^2 + \cos^2(\theta) = 1
\]

\[
\frac{7}{16} + \cos^2(\theta) = 1
\]

\[
\cos^2(\theta) = 1 - \frac{7}{16} = \frac{16}{16} - \frac{7}{16} = \frac{9}{16}
\]

\[
\cos(\theta) = \pm \frac{3}{4}
\]

### Step 2: Determine the sign of \(\cos(\theta)\).
Since \(\sin(\theta)\) is negative, \(\theta\) must be in either the third or fourth quadrant. In the third quadrant, both sine and cosine are negative; in the fourth quadrant, sine is negative and cosine is positive. 

To find \(\cot(\theta)\), we need \(\cos(\theta)\), and its sign depends on the quadrant. However, without further information about the angle, we should consider both cases.

### Step 3: Calculate \(\cot(\theta)\).
\(\cot(\theta)\) is given by:

\[
\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
\]

Substituting the values:

1. If \(\cos(\theta) = \frac{3}{4}\):

\[
\cot(\theta) = \frac{\frac{3}{4}}{-\frac{\sqrt{7}}{4}} = -\frac{3}{\sqrt{7}}
\]

2. If \(\cos(\theta) = -\frac{3}{4}\):

\[
\cot(\theta) = \frac{-\frac{3}{4}}{-\frac{\sqrt{7}}{4}} = \frac{3}{\sqrt{7}}
\]

### Final Answer:
- If \(\theta\) is in the third quadrant, \(\cot(\theta) = \frac{3}{\sqrt{7}}\).
- If \(\theta\) is in the fourth quadrant, \(\cot(\theta) = -\frac{3}{\sqrt{7}}\).

Would you like to know more details or have any further questions?

Here are 5 related questions you might find interesting:

1. How do you find the value of \(\tan(\theta)\) given \(\sin(\theta)\) and \(\cos(\theta)\)?
2. What are the Pythagorean identities in trigonometry?
3. How do the signs of trigonometric functions vary across different quadrants?
4. What is the relationship between cotangent and tangent?
5. How can you use trigonometric identities to solve equations involving \(\sin(\theta)\) and \(\cos(\theta)\)?

**Tip:** Remember, the sign of trigonometric functions depends on the quadrant in which the angle lies.

Learn how to calculate cotangent (cot) with sin(θ) = -(√7)/4 using trigonometric identities and the Pythagorean identity. Step-by-step solution provided with detailed mathematical explanations.

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