Given sin(θ)=-1/2 where tan(θ)<0 find: sin(2θ)=

Given that \( \sin(\theta) = -\frac{1}{2} \) and \( \tan(\theta) < 0 \), we can follow these steps to find \( \sin(2\theta) \).

### Step 1: Identify the quadrant of \( \theta \)
Since \( \sin(\theta) = -\frac{1}{2} \), \( \theta \) must lie in either the third or fourth quadrant because the sine function is negative in both of these quadrants.

Given \( \tan(\theta) < 0 \), we know that the tangent is positive in the third quadrant and negative in the fourth quadrant. Therefore, \( \theta \) lies in the **fourth quadrant**.

### Step 2: Use the identity \( \sin(2\theta) \)
We can use the double-angle identity for sine:
\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]
We already know \( \sin(\theta) = -\frac{1}{2} \), so we now need to find \( \cos(\theta) \).

### Step 3: Find \( \cos(\theta) \)
Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\[
\left(-\frac{1}{2}\right)^2 + \cos^2(\theta) = 1
\]
\[
\frac{1}{4} + \cos^2(\theta) = 1
\]
\[
\cos^2(\theta) = 1 - \frac{1}{4} = \frac{3}{4}
\]
\[
\cos(\theta) = \pm \frac{\sqrt{3}}{2}
\]
Since \( \theta \) is in the fourth quadrant, where cosine is positive, we have:
\[
\cos(\theta) = \frac{\sqrt{3}}{2}
\]

### Step 4: Calculate \( \sin(2\theta) \)
Now we can substitute into the double-angle formula:
\[
\sin(2\theta) = 2 \cdot \left(-\frac{1}{2}\right) \cdot \frac{\sqrt{3}}{2}
\]
\[
\sin(2\theta) = -\frac{\sqrt{3}}{2}
\]

Thus, \( \sin(2\theta) = -\frac{\sqrt{3}}{2} \).

Would you like further details or have any questions?

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Here are five related questions you might find interesting:
1. How do you calculate \( \cos(2\theta) \) using double-angle identities?
2. What happens if \( \tan(\theta) > 0 \) with \( \sin(\theta) = -\frac{1}{2} \)?
3. What are the key quadrant rules for trigonometric functions?
4. Can you derive the identity for \( \cos(2\theta) \) from the Pythagorean identity?
5. How does the angle \( \theta \) in different quadrants affect the signs of trigonometric functions?

**Tip:** Understanding the quadrant where an angle lies is essential to determining the correct signs of trigonometric functions.

Find sin(2θ) given that sin(θ) = -1/2 and tan(θ) < 0. This solution uses trigonometric identities such as the double-angle formula for sine and the Pythagorean identity. Suitable for students in Grades 10-12.

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