https://file.aiquickdraw.com/mathbot/file/1726044900973wr4osp76.jpg

In the image, we have a right-angled triangle with one angle measuring \( 65^\circ \), the hypotenuse of the triangle measuring 12 units, and the base denoted as \( x \).

To find the value of \( x \), we can use trigonometry. Since we have the angle and the hypotenuse, and we are looking for the adjacent side, we can use the cosine function:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

Here, \( \theta = 65^\circ \) and the hypotenuse is 12. So, the equation becomes:

\[
\cos(65^\circ) = \frac{x}{12}
\]

Now, solving for \( x \):

\[
x = 12 \cdot \cos(65^\circ)
\]

Let me calculate that for you.The value of \( x \), the adjacent side of the triangle, is approximately 5.07 units.

Would you like further details or explanations? Here are some related questions you might find interesting:

1. How would you calculate the opposite side of the triangle?
2. What is the sine of \( 65^\circ \), and how can it be used to find the opposite side?
3. How does the Pythagorean theorem apply to this triangle?
4. How would you solve the problem if a different angle were given instead?
5. Can you find the area of this triangle using the sides?

**Tip:** When solving trigonometric problems, always ensure your calculator is in the correct mode (degrees or radians) based on the angle given.

Find the length of the adjacent side (x) in a right triangle where the hypotenuse is 12 units and one angle is 65 degrees.

Learn how to solve for the adjacent side in a right triangle where the hypotenuse is 12 units and the angle is 65 degrees. This problem involves using the cosine function and is suitable for students in Grades 9-11.

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