You are told that cos (theta) = -(3/sqrt(572)). If theta is in the second quadrant, then the exact value of sin (theta) is what?     If theta is in the third quadrant then the exact value of sin(theta) is what?

Given that \(\cos(\theta) = -\frac{3}{\sqrt{572}}\), we need to find the exact value of \(\sin(\theta)\) for both the second and third quadrants.

### Step 1: Use the Pythagorean Identity
The Pythagorean identity states that:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Substitute \(\cos(\theta)\) into the equation:
\[
\sin^2(\theta) + \left(-\frac{3}{\sqrt{572}}\right)^2 = 1
\]

### Step 2: Calculate \(\cos^2(\theta)\)
\[
\left(-\frac{3}{\sqrt{572}}\right)^2 = \frac{9}{572}
\]
So the equation becomes:
\[
\sin^2(\theta) + \frac{9}{572} = 1
\]

### Step 3: Solve for \(\sin^2(\theta)\)
\[
\sin^2(\theta) = 1 - \frac{9}{572}
\]
Simplify the right-hand side:
\[
\sin^2(\theta) = \frac{572}{572} - \frac{9}{572} = \frac{563}{572}
\]

### Step 4: Find \(\sin(\theta)\)
Taking the square root of both sides:
\[
\sin(\theta) = \pm \sqrt{\frac{563}{572}}
\]

Now, we determine the sign of \(\sin(\theta)\) based on the quadrant.

### Case 1: Theta in the Second Quadrant
In the second quadrant, \(\sin(\theta)\) is positive. Therefore:
\[
\sin(\theta) = \sqrt{\frac{563}{572}}
\]

### Case 2: Theta in the Third Quadrant
In the third quadrant, \(\sin(\theta)\) is negative. Therefore:
\[
\sin(\theta) = -\sqrt{\frac{563}{572}}
\]

### Final Answer:
- If \(\theta\) is in the second quadrant, the exact value of \(\sin(\theta)\) is \(\sqrt{\frac{563}{572}}\).
- If \(\theta\) is in the third quadrant, the exact value of \(\sin(\theta)\) is \(-\sqrt{\frac{563}{572}}\).

Do you want details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would you find \(\tan(\theta)\) in both quadrants given \(\cos(\theta)\)?
2. What is the range of values for \(\theta\) in the second and third quadrants?
3. How can you express \(\cos(\theta)\) in terms of \(\sin(\theta)\)?
4. What happens to the signs of trigonometric functions in the fourth quadrant?
5. How does the Pythagorean identity relate to the unit circle?

**Tip:** Understanding the signs of trigonometric functions in each quadrant is key to solving problems like this one effectively.

Learn how to find the exact value of sin(theta) given cos(theta) = -(3/sqrt(572)). Understand the application of the Pythagorean Identity in trigonometry. Suitable for advanced high school students studying trigonometry.

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