Part 1: Finding $\sin(2\theta)$

Given: $\sin(\theta) = \frac{3}{4}$ where $\frac{\pi}{2} \leq \theta \leq \pi$.

We know the identity for $\sin(2\theta)$: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

First, we need to find $\cos(\theta)$.

Finding $\cos(\theta)$

We use the Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$

Substituting $\sin(\theta) = \frac{3}{4}$: $\left(\frac{3}{4}\right)^2 + \cos^2(\theta) = 1$ $\frac{9}{16} + \cos^2(\theta) = 1$ $\cos^2(\theta) = 1 - \frac{9}{16} = \frac{7}{16}$ $\cos(\theta) = \pm \frac{\sqrt{7}}{4}$

Since $\theta$ is in the second quadrant ($\frac{\pi}{2} \leq \theta \leq \pi$), where cosine is negative: $\cos(\theta) = -\frac{\sqrt{7}}{4}$

Now, calculate $\sin(2\theta)$:

$\sin(2\theta) = 2\sin(\theta)\cos(\theta) = 2 \times \frac{3}{4} \times \left(-\frac{\sqrt{7}}{4}\right) = -\frac{3\sqrt{7}}{8}$

Part 2: Finding $\cos(\theta)$ and $\sin(\theta)$

Given: $\cos(2\theta) = -\frac{7}{8}$ and $2\theta$ is in the third quadrant.

The identities for $\cos(2\theta)$ are: $\cos(2\theta) = 2\cos^2(\theta) - 1$ or $\cos(2\theta) = 1 - 2\sin^2(\theta)$

Solving for $\cos(\theta)$:

Using $\cos(2\theta) = 2\cos^2(\theta) - 1$: $-\frac{7}{8} = 2\cos^2(\theta) - 1$ $2\cos^2(\theta) = \frac{1}{8}$ $\cos^2(\theta) = \frac{1}{16}$ $\cos(\theta) = \pm \frac{1}{4}$

Since $\theta$ is in the second quadrant, $\cos(\theta)$ is negative: $\cos(\theta) = -\frac{1}{4}$

Solving for $\sin(\theta)$:

Using the identity: $\sin^2(\theta) + \cos^2(\theta) = 1$ $\sin^2(\theta) + \left(-\frac{1}{4}\right)^2 = 1$ $\sin^2(\theta) + \frac{1}{16} = 1$ $\sin^2(\theta) = \frac{15}{16}$ $\sin(\theta) = \pm \frac{\sqrt{15}}{4}$

Since $\theta$ is in the second quadrant, $\sin(\theta)$ is positive: $\sin(\theta) = \frac{\sqrt{15}}{4}$

Final Answers:

• $\sin(2\theta) = -\frac{3\sqrt{7}}{8}$
• $\cos(\theta) = -\frac{1}{4}$
• $\sin(\theta) = \frac{\sqrt{15}}{4}$

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Would you like more details on any of these steps, or do you have any further questions?

Here are some related questions:

1. What is the significance of the Pythagorean identity?
2. Can you find $\tan(2\theta)$ given $\sin(2\theta)$ and $\cos(2\theta)$?
3. How does the quadrant affect the sign of trigonometric functions?
4. What are the double angle identities for cosine and sine?
5. How would the solution change if $\theta$ were in the fourth quadrant?
6. Can you derive $\sin(2\theta)$ from $\sin(\theta)$ and $\cos(\theta)$ in a different way?
7. What is the importance of the interval $\frac{\pi}{2} \leq \theta \leq \pi$ in this problem?
8. How can we verify the calculated values using a unit circle?

Tip: When working with trigonometric identities, always pay attention to the quadrant to determine the correct sign of the function values.