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We are given that:

\[
\cos A = \frac{2}{3} \quad \text{and} \quad \sin B = \frac{1}{3}
\]
We are asked to show that:

\[
\cos(2A + B) = -\frac{2\sqrt{2}}{27} - \frac{4\sqrt{5}}{27}
\]

### Step-by-Step Solution

#### Step 1: Find \(\sin A\) and \(\cos B\)
We know that:
\[
\sin^2 A + \cos^2 A = 1
\]
Using \(\cos A = \frac{2}{3}\), we calculate \(\sin A\):
\[
\sin^2 A = 1 - \cos^2 A = 1 - \left(\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9}
\]
\[
\sin A = \frac{\sqrt{5}}{3}
\]

Next, using \(\sin B = \frac{1}{3}\), we can calculate \(\cos B\):
\[
\sin^2 B + \cos^2 B = 1
\]
\[
\cos^2 B = 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9}
\]
\[
\cos B = \frac{2\sqrt{2}}{3}
\]

#### Step 2: Use the formula for \(\cos(2A + B)\)
We use the angle sum identity:
\[
\cos(2A + B) = \cos 2A \cos B - \sin 2A \sin B
\]

We now need to find \(\cos 2A\) and \(\sin 2A\).

#### Step 3: Find \(\cos 2A\) and \(\sin 2A\)
We use the double angle identities:
\[
\cos 2A = 2\cos^2 A - 1 = 2\left(\frac{2}{3}\right)^2 - 1 = 2\left(\frac{4}{9}\right) - 1 = \frac{8}{9} - 1 = -\frac{1}{9}
\]
\[
\sin 2A = 2\sin A \cos A = 2\left(\frac{\sqrt{5}}{3}\right)\left(\frac{2}{3}\right) = \frac{4\sqrt{5}}{9}
\]

#### Step 4: Substitute into \(\cos(2A + B)\)
Now, substituting everything into the identity for \(\cos(2A + B)\):
\[
\cos(2A + B) = \left(-\frac{1}{9}\right)\left(\frac{2\sqrt{2}}{3}\right) - \left(\frac{4\sqrt{5}}{9}\right)\left(\frac{1}{3}\right)
\]
\[
= -\frac{2\sqrt{2}}{27} - \frac{4\sqrt{5}}{27}
\]

Thus, we have shown that:
\[
\cos(2A + B) = -\frac{2\sqrt{2}}{27} - \frac{4\sqrt{5}}{27}
\]

Let me know if you need further clarifications or additional details. 

### Related Questions:
1. What is the formula for double angles, and how does it apply in other trigonometric identities?
2. How can we generalize this process for other combinations of angles in trigonometry?
3. What other trigonometric identities involve summing angles, and how are they derived?
4. How can you use a calculator to verify trigonometric expressions like this one?
5. What is the geometrical interpretation of trigonometric identities for acute angles?

### Tip:
When solving problems involving trigonometric identities, always write down the basic identities and apply them step-by-step to avoid errors.

A and B are acute angles such that cos(A) = 2/3 and sin(B) = 1/3. Without finding the size of A or B, show that cos(2A + B) = -(2√2)/27 - (4√5)/27.

This page provides a step-by-step solution to a trigonometry problem involving acute angles A and B, where cos(A) = 2/3 and sin(B) = 1/3. Learn how to use angle sum and double angle formulas to show that cos(2A + B) = -(2√2)/27 - (4√5)/27, without finding the size of the angles.

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