if cot() = 3/5 and theta is in quadrant 3, find sin theta and sec theta

We are given that \(\cot(\theta) = \frac{3}{5}\) and \(\theta\) is in Quadrant III. To find \(\sin(\theta)\) and \(\sec(\theta)\), we can proceed step by step.

### Step 1: Relationship between cotangent and tangent
\[
\cot(\theta) = \frac{1}{\tan(\theta)} \implies \tan(\theta) = \frac{5}{3}
\]

### Step 2: Using the Pythagorean identity to find \(\sin(\theta)\)
Recall the identity:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x}
\]
So, \(y = 5\) and \(x = -3\) (since \(\theta\) is in Quadrant III where both \(x\) and \(y\) are negative).

Now, use the Pythagorean theorem to find the hypotenuse \(r\):
\[
r = \sqrt{x^2 + y^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}
\]

### Step 3: Find \(\sin(\theta)\)
Since \(\sin(\theta) = \frac{y}{r}\) and \(y = -5\) (negative in Quadrant III):
\[
\sin(\theta) = \frac{-5}{\sqrt{34}}
\]
To rationalize the denominator:
\[
\sin(\theta) = \frac{-5\sqrt{34}}{34}
\]

### Step 4: Find \(\sec(\theta)\)
We know that \(\sec(\theta) = \frac{1}{\cos(\theta)}\), and \(\cos(\theta) = \frac{x}{r}\):
\[
\cos(\theta) = \frac{-3}{\sqrt{34}}
\]
Rationalizing the denominator:
\[
\cos(\theta) = \frac{-3\sqrt{34}}{34}
\]
Thus,
\[
\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{34}{-3\sqrt{34}} = \frac{-\sqrt{34}}{3}
\]

### Final Answer:
\[
\sin(\theta) = \frac{-5\sqrt{34}}{34}, \quad \sec(\theta) = \frac{-\sqrt{34}}{3}
\]

Would you like further explanation on any step or more details?

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**Related Questions**:
1. How is the sign of \(\sin(\theta)\) determined in different quadrants?
2. What are the trigonometric identities related to tangent and cotangent?
3. How do you derive the secant function from cosine?
4. Why do we rationalize denominators in trigonometry?
5. How do the values of trigonometric functions change in Quadrant IV?

**Tip**: Always keep track of signs in trigonometry by considering which quadrant the angle lies in!

if cot(θ) = 3/5 and theta is in quadrant 3, find sin(θ) and sec(θ)

Solve for sin(θ) and sec(θ) when cot(θ) = 3/5 and θ is in Quadrant III. This problem covers trigonometric concepts like cotangent, sine, secant functions, and the Pythagorean theorem, with a step-by-step solution. Suitable for Grades 10-12 students.

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