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For exercise 5:

Given:
\[
\tan x = \frac{3}{4}, \quad \sec y = \frac{13}{5}
\]

First, calculate \(\cos y\) and \(\sin y\):
\[
\cos y = \frac{1}{\sec y} = \frac{5}{13}
\]
\[
\sin y = \sqrt{1 - \cos^2 y} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13}
\]

Next, calculate \(\sin x\):
\[
\tan x = \frac{\sin x}{\cos x} \implies \cos x = \frac{1}{\sqrt{1 + \tan^2 x}} = \frac{1}{\sqrt{1 + \left(\frac{3}{4}\right)^2}} = \frac{1}{\sqrt{1 + \frac{9}{16}}} = \frac{1}{\sqrt{\frac{25}{16}}} = \frac{4}{5}
\]
\[
\sin x = \tan x \cdot \cos x = \frac{3}{4} \cdot \frac{4}{5} = \frac{3}{5}
\]

Now, use the sine addition formula:
\[
\sin(x + y) = \sin x \cos y + \cos x \sin y
\]
\[
\sin(x + y) = \frac{3}{5} \cdot \frac{5}{13} + \frac{4}{5} \cdot \frac{12}{13} = \frac{15}{65} + \frac{48}{65} = \frac{63}{65}
\]

Thus, \(\sin(x + y) = \frac{63}{65} \approx 0.9692\).

For exercise 7, we need to calculate \(\tan x\) using the given right triangle.

First, observe the figure and the given values:
\[
\tan 30^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{2}{\sqrt{4^2 - 2^2}} = \frac{2}{\sqrt{16 - 4}} = \frac{2}{\sqrt{12}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]

Given that:
\[
\tan x = \frac{\text{opposite}}{\text{adjacent}}
\]

For the small triangle within the large triangle:
\[
\tan x = \frac{h}{2}
\]

Using the values from the large triangle:
\[
h = 4 \sin 30^\circ = 4 \cdot \frac{1}{2} = 2
\]
So:
\[
\tan x = \frac{2}{2} = 1
\]

Thus, \(\tan x = 1\).

If you have any questions or need further details, please let me know!

Here are five additional questions you can ask:
1. How do I find the value of \(\cos(x + y)\)?
2. Can you solve the remaining exercises in the image?
3. How do I use the Pythagorean theorem in trigonometry?
4. What are some applications of trigonometric identities?
5. How can I simplify trigonometric expressions?

**Tip:** Always verify your trigonometric values using the unit circle or right triangle properties to ensure accuracy.

Explore detailed solutions for exercises 5 and 7 in trigonometry, including calculations for sine and tangent functions. Understand key concepts like the sine addition formula and trigonometric identities. Ideal for high school students.

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