https://file.aiquickdraw.com/mathbot/file/1733187367880lz4ofr4d.jpg

Let’s break down each expression using the appropriate compound angle formulas.

### Compound Angle Formulas
1. **Sine Addition Formula**: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)
2. **Cosine Addition Formula**: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)
3. **Tangent Addition Formula**: \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\)

Now, solving each subpart:

---

### a) \(\sin(\pi + x)\)
Using the sine addition formula:
\[
\sin(\pi + x) = \sin(\pi)\cos(x) + \cos(\pi)\sin(x)
\]
\(\sin(\pi) = 0\) and \(\cos(\pi) = -1\), so:
\[
\sin(\pi + x) = 0 \cdot \cos(x) + (-1) \cdot \sin(x) = -\sin(x)
\]

---

### b) \(\cos\left(x + \frac{3\pi}{2}\right)\)
Using the cosine addition formula:
\[
\cos\left(x + \frac{3\pi}{2}\right) = \cos(x)\cos\left(\frac{3\pi}{2}\right) - \sin(x)\sin\left(\frac{3\pi}{2}\right)
\]
\(\cos\left(\frac{3\pi}{2}\right) = 0\) and \(\sin\left(\frac{3\pi}{2}\right) = -1\), so:
\[
\cos\left(x + \frac{3\pi}{2}\right) = \cos(x) \cdot 0 - \sin(x) \cdot (-1) = \sin(x)
\]

---

### c) \(\sin(x - \pi)\)
Using the sine subtraction formula:
\[
\sin(x - \pi) = \sin(x)\cos(\pi) - \cos(x)\sin(\pi)
\]
\(\cos(\pi) = -1\) and \(\sin(\pi) = 0\), so:
\[
\sin(x - \pi) = \sin(x) \cdot (-1) - \cos(x) \cdot 0 = -\sin(x)
\]

---

### d) \(\tan(x + \pi)\)
Using the tangent addition formula:
\[
\tan(x + \pi) = \frac{\tan(x) + \tan(\pi)}{1 - \tan(x)\tan(\pi)}
\]
\(\tan(\pi) = 0\), so:
\[
\tan(x + \pi) = \frac{\tan(x) + 0}{1 - \tan(x) \cdot 0} = \tan(x)
\]
However, for \(\tan(x + \pi)\), \(\tan(x)\) also flips its sign due to periodicity:
\[
\tan(x + \pi) = -\tan(x)
\]

---

### e) \(\tan(2\pi - x)\)
Using the tangent subtraction formula:
\[
\tan(2\pi - x) = \frac{\tan(2\pi) - \tan(x)}{1 + \tan(2\pi)\tan(x)}
\]
\(\tan(2\pi) = 0\), so:
\[
\tan(2\pi - x) = \frac{0 - \tan(x)}{1 + 0 \cdot \tan(x)} = -\tan(x)
\]

---

### Final Results
1. **a)** \(-\sin(x)\)
2. **b)** \(\sin(x)\)
3. **c)** \(-\sin(x)\)
4. **d)** \(-\tan(x)\)
5. **e)** \(-\tan(x)\)

---

Would you like a more detailed explanation for any of these?  
Here are 5 related questions to expand your understanding:
1. How does the periodicity of trigonometric functions affect their compound angle formulas?
2. What happens to the values of \(\sin\), \(\cos\), and \(\tan\) at specific angles like \( \pi \) or \( \frac{3\pi}{2} \)?
3. Why does the tangent function change its sign when shifted by \(\pi\)?
4. Can you derive the compound angle formulas from the unit circle?
5. What are the applications of these formulas in solving real-world problems?

**Tip**: Always keep track of the periodicity of trigonometric functions—it helps simplify seemingly complex expressions!

Use the appropriate compound angle formula to create an equivalent expression for the given trigonometric functions.

Learn how to simplify trigonometric expressions such as sin(π + x), cos(x + 3π/2), and tan(2π - x) using compound angle formulas. Understand the periodic properties of sine, cosine, and tangent. This guide is ideal for advanced high school students in Grades 11-12.

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