The problem asks to match trigonometric expressions with their corresponding non-trigonometric functions. Let's go step-by-step and solve it.

Given expressions:

1. $\sin(\theta) = \frac{x}{2}$ — Find $\tan(\theta)$.
2. $\sin(\theta) = \frac{x}{2}$ — Find $\cos(\theta)$.
3. $\sin(\theta) = \frac{x}{2}$ — Find $\sin(2\theta)$.
4. $\tan(\theta) = \frac{x}{\sqrt{4 - x^2}}$ — Find $\sin(\theta)$.
5. $\tan(\theta) = \frac{x}{2}$ — Find $\cos(\theta)$.

Using Trigonometric Identities:

• Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$.
• Double angle identity for sine: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.

Step-by-step matching:

1. A. $\sin(\theta) = \frac{x}{2}$ — Find $\tan(\theta)$: Using the identity $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ and solving for $\cos(\theta)$ from $\sin^2(\theta) + \cos^2(\theta) = 1$, we have: $\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left( \frac{x}{2} \right)^2} = \sqrt{1 - \frac{x^2}{4}} = \frac{\sqrt{4 - x^2}}{2}$ So, $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{x}{2}}{\frac{\sqrt{4 - x^2}}{2}} = \frac{x}{\sqrt{4 - x^2}}.$ This matches option 4.

2. B. $\sin(\theta) = \frac{x}{2}$ — Find $\cos(\theta)$: We already solved $\cos(\theta)$ in the previous step: $\cos(\theta) = \frac{\sqrt{4 - x^2}}{2}.$ This matches option 3.

3. C. $\sin(\theta) = \frac{x}{2}$ — Find $\sin(2\theta)$: Using the double angle identity $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, we substitute: $\sin(2\theta) = 2 \cdot \frac{x}{2} \cdot \frac{\sqrt{4 - x^2}}{2} = \frac{x\sqrt{4 - x^2}}{2}.$ This matches option 5.

4. D. $\tan(\theta) = \frac{x}{\sqrt{4 - x^2}}$ — Find $\sin(\theta)$: We use the identity $\sin(\theta) = \frac{\tan(\theta)}{\sqrt{1 + \tan^2(\theta)}}$. Substituting $\tan(\theta) = \frac{x}{\sqrt{4 - x^2}}$: $\sin(\theta) = \frac{\frac{x}{\sqrt{4 - x^2}}}{\sqrt{1 + \left( \frac{x}{\sqrt{4 - x^2}} \right)^2}} = \frac{\frac{x}{\sqrt{4 - x^2}}}{\sqrt{ \frac{4}{4 - x^2}}} = \frac{x}{2}.$ This matches option 1.

5. E. $\tan(\theta) = \frac{x}{2}$ — Find $\cos(\theta)$: We know that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Using the Pythagorean identity for $\cos(\theta)$, we have: $\cos(\theta) = \frac{\sqrt{4 - x^2}}{2}.$ This matches option 2.

Final Answers:

1. $A$ matches 4.
2. $B$ matches 3.
3. $C$ matches 5.
4. $D$ matches 1.
5. $E$ matches 2.

Let me know if you'd like more details or have any questions! Here are 5 related questions to practice further:

1. How do you derive the double-angle identity for sine?
2. How can you solve for $\sin(\theta)$ given a value for $\tan(\theta)$?
3. What is the difference between using the Pythagorean identity and the double-angle identity?
4. How do you find $\cos(2\theta)$ using the identities?
5. What happens to $\sin(\theta)$ and $\cos(\theta)$ when $\theta = 90^\circ$?

Tip: When working with trigonometric identities, drawing a triangle and labeling the sides can be helpful for visualizing the relationships!