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To solve for \( x \) in this right triangle, we can use trigonometric relationships since the angle \( 67^\circ \) and the hypotenuse (13) are given.

### Step-by-step process:

1. **Identify the correct trigonometric function:**
   - The side \( x \) is adjacent to the given angle \( 67^\circ \), and the hypotenuse is 13.
   - The cosine function relates the adjacent side and the hypotenuse: 
     \[
     \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
     \]
     Here, \( \theta = 67^\circ \), adjacent side is \( x \), and hypotenuse is 13.

2. **Set up the equation:**
   \[
   \cos(67^\circ) = \frac{x}{13}
   \]

3. **Solve for \( x \):**
   \[
   x = 13 \times \cos(67^\circ)
   \]
   
4. **Calculate the value:**

I'll calculate the value of \( x \).The length of \( x \) is approximately \( 5.08 \) units, rounded to the hundredths place.

Would you like further details or have any additional questions?

Here are 5 related questions to extend this topic:
1. How would you find the length of the other side of the triangle?
2. What is the sine of the given \( 67^\circ \) angle?
3. How does the tangent function relate to this triangle?
4. What would the angle be if \( x \) and the hypotenuse were switched?
5. How can you verify this answer using the Pythagorean theorem?

**Tip:** Always ensure you're working with the correct trigonometric function for the given triangle sides!

For the right triangle below, find the length of x. Round to the hundredths. (2 decimal places) The hypotenuse is 13 and the given angle is 67 degrees.

Learn how to solve for the length of the side x in a right triangle with a given hypotenuse of 13 and an angle of 67°. Using cosine trigonometric functions, the solution is calculated and rounded to two decimal places.

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