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Let's go through the three parts of the problem one by one.

### 23. a) 
You are asked to calculate the value of the expression:

\[
E = \frac{3\sin x + 2\cos x}{3\cos x + 2\sin x}
\]

given that \(\tan x = 3\).

To solve this:

1. Use the identity \(\tan x = \frac{\sin x}{\cos x}\) to find \(\sin x\) and \(\cos x\).
2. Plug these values into the expression for \(E\) and simplify.

### 23. b)
You need to evaluate the expression:

\[
E = (\sin x - \cos x)^2 + \sin 2x
\]

for \(x = \frac{\pi}{3}\).

1. Substitute \(x = \frac{\pi}{3}\) into \(\sin x\), \(\cos x\), and \(\sin 2x\).
2. Simplify the expression.

### 23. c)
You need to calculate the expression:

\[
E = (\sin x - \cos y)^2 + (\sin y - \cos x)^2
\]

given that \(x + y = \frac{\pi}{6}\).

1. Use the relation \(x + y = \frac{\pi}{6}\) to express one variable in terms of the other (for example, \(y = \frac{\pi}{6} - x\)).
2. Substitute these into the trigonometric terms and simplify.

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I'll begin by calculating part a first, and then move on to the other parts. Let me do the math for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can walk you through the calculations manually or provide further clarifications on any part of the problem. Let me know how you'd like to proceed!

Here are some related questions for further understanding:
1. How do you derive \(\sin x\) and \(\cos x\) from \(\tan x = 3\)?
2. What are the key trigonometric identities to simplify these types of expressions?
3. How do you evaluate trigonometric functions at common angles like \(\frac{\pi}{3}\)?
4. What are the steps for solving expressions involving \(\sin 2x\)?
5. How does the relationship \(x + y = \frac{\pi}{6}\) help simplify the expression in part c?

**Tip:** Remember that simplifying complex trigonometric expressions often involves using key identities like Pythagoras and angle sum formulas!

Solve the following trigonometric expressions:
1. Calculate the value of the expression E = (3sin x + 2cos x) / (3cos x + 2sin x) for tan(x) = 3.
2. Find the value of the expression E = (sin x - cos x)^2 + sin(2x) for x = π/3.
3. Calculate the expression E = (sin x - cos y)^2 + (sin y - cos x)^2 for x + y = π/6.

This problem involves solving trigonometric expressions using key identities for sine, cosine, and tangent. It includes calculating expressions for given values of x and y, using identities like sin(2x) and tan(x). Suitable for high school students in Grades 10-12.

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