Given cosine, theta, equals, minus, one halfcosθ=−
2
1
​
and angle thetaθ is in Quadrant II, what is the exact value of sine, thetasinθ in simplest form? Simplify all radicals if needed.

We are tasked with finding \(\sin\theta\) given that \(\cos\theta = -\frac{1}{2}\) and that \(\theta\) is in Quadrant II.

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### Step 1: Recall the Pythagorean Identity
The Pythagorean identity is:
\[
\sin^2\theta + \cos^2\theta = 1
\]

### Step 2: Substitute the value of \(\cos\theta\)
\[
\sin^2\theta + \left(-\frac{1}{2}\right)^2 = 1
\]
Simplify:
\[
\sin^2\theta + \frac{1}{4} = 1
\]

### Step 3: Solve for \(\sin^2\theta\)
\[
\sin^2\theta = 1 - \frac{1}{4}
\]
\[
\sin^2\theta = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}
\]

### Step 4: Find \(\sin\theta\)
Take the square root of both sides:
\[
\sin\theta = \pm\sqrt{\frac{3}{4}}
\]
Simplify the square root:
\[
\sin\theta = \pm\frac{\sqrt{3}}{2}
\]

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### Step 5: Determine the correct sign for \(\sin\theta\)
Since \(\theta\) is in Quadrant II, where sine is positive:
\[
\sin\theta = \frac{\sqrt{3}}{2}
\]

---

### Final Answer:
\[
\sin\theta = \frac{\sqrt{3}}{2}
\]

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Would you like further clarification on any step?  

Here are 5 related questions for deeper understanding:  
1. What is the significance of the Pythagorean identity in trigonometry?  
2. Why is sine positive in Quadrant II?  
3. How would the problem change if \(\theta\) were in Quadrant III?  
4. What are the other trigonometric ratios of \(\theta\) in this scenario?  
5. How can this result be verified on the unit circle?  

**Tip:** Always consider the quadrant when determining the sign of a trigonometric function!

Given cosine, theta, equals, minus, one halfcosθ=−1/2 and angle thetaθ is in Quadrant II, what is the exact value of sine, thetasinθ in simplest form? Simplify all radicals if needed.

Learn how to determine the exact value of sine (θ) given that cos(θ) = -1/2 and θ is in Quadrant II. This step-by-step solution uses the Pythagorean identity and trigonometric sign rules for Quadrant II. Suitable for high school students.

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