https://file.aiquickdraw.com/mathbot/file/1725196112904wzv455ws.jpg

The problem asks for the length of side \( AC \) in the right triangle shown in the image, with a given angle \( \angle A = 65^\circ \) and side \( AB = 7 \) units.

### Step 1: Analyzing the Triangle
In this right triangle:
- \( AB \) is the side adjacent to \( \angle A \).
- \( AC \) is the hypotenuse of the triangle.
- \( \angle A \) is given as \( 65^\circ \).

To find the length of \( AC \), we can use the cosine function, which relates the adjacent side to the hypotenuse.

### Step 2: Using the Cosine Function
The cosine of an angle in a right triangle is given by:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
For \( \angle A = 65^\circ \):
\[
\cos(65^\circ) = \frac{AB}{AC}
\]
Substituting the known value:
\[
\cos(65^\circ) = \frac{7}{AC}
\]

### Step 3: Solving for \( AC \)
Rearranging the equation to solve for \( AC \):
\[
AC = \frac{7}{\cos(65^\circ)}
\]

### Step 4: Calculation
Now, we calculate \( AC \) using the cosine of \( 65^\circ \):

Let me calculate this for you.The length of side \( AC \) is approximately **16.56** units, rounded to the nearest hundredth.

If you have any further questions or need additional clarification, feel free to ask!

### Related Questions:
1. What is the length of side \( BC \)?
2. How do you calculate \( \sin(65^\circ) \) using the triangle?
3. What is the area of this right triangle?
4. Can you find the measure of \( \angle B \)?
5. How would you use the Pythagorean theorem to verify the lengths?

### Tip:
When working with trigonometric functions, always ensure your calculator is in the correct mode (degrees or radians) depending on the angle units provided.

Learn how to calculate the length of side AC in a right triangle given angle A = 65 degrees and side AB = 7 units using trigonometric functions. Detailed step-by-step solution included.

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