https://file.aiquickdraw.com/mathbot/file/17285833412640qyhg8im.jpg

Let's tackle the two problems shown in the image:

### Problem 8:
We are asked to find the value of \(\sin(x + y)\), given:
- \(0 < x < \frac{\pi}{2}\), \(\frac{\pi}{2} < y < \pi\),
- \(\tan(x) = 2\),
- \(\sin(y) = \frac{8}{17}\).

We can use the sine addition formula:
\[
\sin(x + y) = \sin(x) \cos(y) + \cos(x) \sin(y).
\]

#### Step 1: Find \(\sin(x)\) and \(\cos(x)\) from \(\tan(x) = 2\).
Using the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), we know:
\[
\frac{\sin(x)}{\cos(x)} = 2 \quad \Rightarrow \quad \sin(x) = 2 \cos(x).
\]
Now, use the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\):
\[
(2\cos(x))^2 + \cos^2(x) = 1 \quad \Rightarrow \quad 4\cos^2(x) + \cos^2(x) = 1 \quad \Rightarrow \quad 5\cos^2(x) = 1.
\]
Thus:
\[
\cos(x) = \frac{1}{\sqrt{5}}, \quad \sin(x) = 2\cos(x) = \frac{2}{\sqrt{5}}.
\]

#### Step 2: Find \(\cos(y)\) from \(\sin(y) = \frac{8}{17}\).
Using \(\sin^2(y) + \cos^2(y) = 1\):
\[
\left(\frac{8}{17}\right)^2 + \cos^2(y) = 1 \quad \Rightarrow \quad \frac{64}{289} + \cos^2(y) = 1 \quad \Rightarrow \quad \cos^2(y) = \frac{225}{289}.
\]
Thus:
\[
\cos(y) = -\frac{15}{17}, \quad (\text{since } y \in \left( \frac{\pi}{2}, \pi \right) \text{, where cosine is negative}).
\]

#### Step 3: Substitute into the sine addition formula.
Now we substitute the values:
\[
\sin(x + y) = \frac{2}{\sqrt{5}} \cdot -\frac{15}{17} + \frac{1}{\sqrt{5}} \cdot \frac{8}{17} = -\frac{30}{17\sqrt{5}} + \frac{8}{17\sqrt{5}} = \frac{-30 + 8}{17\sqrt{5}} = \frac{-22}{17\sqrt{5}}.
\]
Rationalizing the denominator:
\[
\sin(x + y) = \frac{-22\sqrt{5}}{85}.
\]
Thus, the correct answer is:
\[
\boxed{A} \quad \left(\frac{-22\sqrt{5}}{85}\right).
\]

---

### Problem 16:
We are asked to find the value of \((\cos(3x) + \sin(3x))^2\), given \(\sin(6x) = \frac{1}{4}\).

#### Step 1: Use the identity \(\sin(6x) = 2 \sin(3x) \cos(3x)\).
We are given \(\sin(6x) = \frac{1}{4}\), so:
\[
2 \sin(3x) \cos(3x) = \frac{1}{4} \quad \Rightarrow \quad \sin(3x) \cos(3x) = \frac{1}{8}.
\]

#### Step 2: Apply the expansion.
We need to find \((\cos(3x) + \sin(3x))^2\). Expanding this gives:
\[
(\cos(3x) + \sin(3x))^2 = \cos^2(3x) + 2 \cos(3x) \sin(3x) + \sin^2(3x).
\]
Using the identity \(\cos^2(3x) + \sin^2(3x) = 1\) and substituting \(\sin(3x) \cos(3x) = \frac{1}{8}\), we get:
\[
(\cos(3x) + \sin(3x))^2 = 1 + 2 \times \frac{1}{8} = 1 + \frac{1}{4} = \frac{5}{4}.
\]
Thus, the correct answer is:
\[
\boxed{B} \quad \left(\frac{5}{4}\right).
\]

---

Would you like any further details or clarifications? 

Here are five related questions to help you deepen your understanding:
1. How would you derive \(\tan(x)\) from \(\sin(x)\) and \(\cos(x)\)?
2. What other trigonometric identities involve \(\sin(x + y)\)?
3. How does the quadrant of an angle affect the sign of trigonometric functions?
4. What is the relationship between \(\sin(2x)\) and \(\sin(x)\)?
5. How can you verify if an angle belongs to a particular quadrant based on the sine or cosine values?

**Tip**: When solving trigonometric problems involving multiple angles, it’s helpful to remember the basic identities and how the sign of each function changes in different quadrants.

Suppose that 0 < x < π/2, π/2 < y < π, tan(x) = 2 and sin(y) = 8/17. Find the value of sin(x + y). Given x is an acute angle with sin(6x) = 1/4, find the value of (cos(3x) + sin(3x))^2.

Learn how to solve trigonometric problems involving the sine addition formula and double angle identities. We solve for sin(x + y) given tan(x) = 2 and sin(y) = 8/17, and find (cos(3x) + sin(3x))^2 using the given sin(6x) = 1/4. Suitable for high school students in Grades 10-12.

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