Use identities to find the following. Do not use a calculator.
​(a) sine 2 theta
​(b) cosine 2 theta
sine theta equals negative StartFraction StartRoot 7 EndRoot Over 9 EndFraction
 
and cosine theta
greater than
0
Question content area bottom
Part 1
sine 2 theta
equalsenter your response here
​(Simplify your​ answer, including any radicals. Use integers or fractions for any numbers in the​ expression.)
Part 2
cosine 2 theta
equalsenter your response here
​(Type an integer or a simplified​ fraction.)

To solve for \( \sin 2\theta \) and \( \cos 2\theta \), we can use the double-angle identities:

1. \( \sin 2\theta = 2 \sin \theta \cos \theta \)
2. \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

Given:
- \( \sin \theta = -\frac{\sqrt{7}}{9} \)
- \( \cos \theta > 0 \), which places \( \theta \) in the fourth quadrant.

### Step 1: Find \( \cos \theta \) using the Pythagorean identity

Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we have:
\[
\cos^2 \theta = 1 - \sin^2 \theta
\]
Substituting \( \sin \theta = -\frac{\sqrt{7}}{9} \):
\[
\cos^2 \theta = 1 - \left(-\frac{\sqrt{7}}{9}\right)^2
\]
\[
\cos^2 \theta = 1 - \frac{7}{81} = \frac{81 - 7}{81} = \frac{74}{81}
\]
Since \( \cos \theta > 0 \), we take the positive square root:
\[
\cos \theta = \sqrt{\frac{74}{81}} = \frac{\sqrt{74}}{9}
\]

### Part 1: Find \( \sin 2\theta \)

Using \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[
\sin 2\theta = 2 \left(-\frac{\sqrt{7}}{9}\right) \left(\frac{\sqrt{74}}{9}\right)
\]
\[
\sin 2\theta = -\frac{2 \sqrt{7} \sqrt{74}}{81} = -\frac{2 \sqrt{518}}{81}
\]

So:
\[
\sin 2\theta = -\frac{2 \sqrt{518}}{81}
\]

### Part 2: Find \( \cos 2\theta \)

Using \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \):
\[
\cos 2\theta = \frac{74}{81} - \frac{7}{81}
\]
\[
\cos 2\theta = \frac{74 - 7}{81} = \frac{67}{81}
\]

### Final Answers

1. \( \sin 2\theta = -\frac{2 \sqrt{518}}{81} \)
2. \( \cos 2\theta = \frac{67}{81} \)

Would you like further details on this solution, or have any questions?

---

1. How is the double-angle identity for sine applied in other scenarios?
2. Can you show a step-by-step derivation of the Pythagorean identity?
3. What would be different if \( \cos \theta < 0 \)?
4. How does quadrant affect the sign of trigonometric functions?
5. How can you simplify expressions involving radicals?

**Tip:** Double-check your quadrant to ensure correct signs in trigonometric identities.

Use identities to find the following. Do not use a calculator.
(a) sine 2 theta
(b) cosine 2 theta
sine theta equals negative StartFraction StartRoot 7 EndRoot Over 9 EndFraction and cosine theta greater than 0.

Find sin(2θ) and cos(2θ) without a calculator by applying trigonometric identities. This solution involves using the double-angle identities and the Pythagorean identity, with sin(θ) given as -√7/9 and cos(θ) > 0. Detailed steps for students in Grades 10-12.

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