Cot theta=-2/8,cos theta<0, find sin theta

We are given that:

\[
\cot(\theta) = -\frac{2}{8} = -\frac{1}{4}
\]

and

\[
\cos(\theta) < 0.
\]

To find \(\sin(\theta)\), let's proceed step by step:

### Step 1: Find \(\tan(\theta)\)
We know that:

\[
\cot(\theta) = \frac{1}{\tan(\theta)}
\]

So,

\[
\tan(\theta) = -4.
\]

### Step 2: Use the identity involving \(\sin(\theta)\) and \(\cos(\theta)\)
The identity for \(\tan(\theta)\) is:

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

Thus, we have:

\[
-4 = \frac{\sin(\theta)}{\cos(\theta)}.
\]

This gives us the relation:

\[
\sin(\theta) = -4 \cos(\theta).
\]

### Step 3: Apply the Pythagorean identity
Using the fundamental identity:

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

we substitute \(\sin(\theta) = -4 \cos(\theta)\) into this equation:

\[
(-4 \cos(\theta))^2 + \cos^2(\theta) = 1,
\]

which simplifies to:

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

\[
17 \cos^2(\theta) = 1,
\]

\[
\cos^2(\theta) = \frac{1}{17}.
\]

### Step 4: Solve for \(\cos(\theta)\)
Since we are told that \(\cos(\theta) < 0\), we take the negative square root:

\[
\cos(\theta) = -\frac{1}{\sqrt{17}}.
\]

### Step 5: Find \(\sin(\theta)\)
Now, substitute \(\cos(\theta)\) into the equation \(\sin(\theta) = -4 \cos(\theta)\):

\[
\sin(\theta) = -4 \left(-\frac{1}{\sqrt{17}}\right),
\]

\[
\sin(\theta) = \frac{4}{\sqrt{17}}.
\]

Thus, the value of \(\sin(\theta)\) is:

\[
\sin(\theta) = \frac{4}{\sqrt{17}}.
\]

---

Would you like more details on any of these steps or have any questions? Here are some related questions for practice:

1. If \(\sin(\theta) = \frac{4}{\sqrt{17}}\), find \(\cos(\theta)\).
2. Given \(\tan(\theta) = -4\), what is the value of \(\sec(\theta)\)?
3. If \(\sin(\theta) = \frac{4}{\sqrt{17}}\), what is the value of \(\csc(\theta)\)?
4. Find the exact value of \(\sin(2\theta)\) given \(\cot(\theta) = -\frac{1}{4}\).
5. What quadrant is \(\theta\) located in if \(\cos(\theta) < 0\) and \(\cot(\theta) < 0\)?

**Tip:** Always determine the correct quadrant based on given information to ensure the correct sign of trigonometric functions.

Cot theta = -2/8, cos theta < 0, find sin theta

Discover how to find sin θ when cot(θ) = -1/4 and cos(θ) < 0 with a step-by-step trigonometric solution. This problem applies the Pythagorean identity and trigonometric formulas, suitable for students in Grades 10-12.

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