https://file.aiquickdraw.com/mathbot/file/1724122323828pprauhg9.jpg

The problem in the image involves solving for the maximum value of a trigonometric expression under a given constraint. Let's break it down step by step:

### Given:
\[ z = \cos^2 \theta + \sin^2 \theta + \frac{5}{3} \sin 2\theta \]

### Constraint:
\[ z \geq \frac{1}{2} \]

### Goal:
Determine the maximum possible value of \( z \).

### Step 1: Simplify the Expression
We know that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
Therefore, the expression for \( z \) simplifies to:
\[ z = 1 + \frac{5}{3} \sin 2\theta \]

### Step 2: Consider the Range of \( z \)
Since \( \sin 2\theta \) varies between -1 and 1, the term \( \frac{5}{3} \sin 2\theta \) will vary between \( -\frac{5}{3} \) and \( \frac{5}{3} \). Therefore:
\[ z \in \left[ 1 - \frac{5}{3}, 1 + \frac{5}{3} \right] \]
\[ z \in \left[ -\frac{2}{3}, \frac{8}{3} \right] \]

### Step 3: Apply the Constraint
We are given that \( z \geq \frac{1}{2} \). Therefore, we need to find the maximum value of \( z \) subject to this constraint.

The maximum possible value of \( z \) is \( \frac{8}{3} \) (which occurs when \( \sin 2\theta = 1 \)).

Since \( \frac{8}{3} \) is greater than \( \frac{1}{2} \), the constraint does not affect the maximum value. 

### Conclusion:
The maximum possible value of \( z \) is \( \frac{8}{3} \).

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. How do you determine the range of a trigonometric expression like \( \frac{5}{3} \sin 2\theta \)?
2. Why is the identity \( \cos^2 \theta + \sin^2 \theta = 1 \) crucial in simplifying the expression for \( z \)?
3. What is the significance of the constraint \( z \geq \frac{1}{2} \) in this problem?
4. How would the maximum value of \( z \) change if the coefficient of \( \sin 2\theta \) were different?
5. Can you derive the general formula for the maximum and minimum values of \( z \) when the expression includes a sine or cosine function?

**Tip:** When solving trigonometric problems, always consider the fundamental identities like \( \sin^2 \theta + \cos^2 \theta = 1 \) as they simplify the expressions and make the problem more manageable.

Explore how to find the maximum value of a trigonometric expression involving cos^2 θ, sin^2 θ, and sin 2θ. Learn step-by-step methods and crucial trigonometric identities. Suitable for advanced high school or college students.

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