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The question involves finding the length of side \( z \) in a right triangle. The given information is as follows:
- One angle measures \( 54^\circ \),
- The adjacent side to this angle is 8 cm,
- The side \( z \) is the opposite side of the triangle, corresponding to the given angle.

To solve for \( z \), we can use the trigonometric function tangent (\(\tan\)) because we have the opposite side and adjacent side.

The formula for tangent is:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Substituting the known values:

\[
\tan(54^\circ) = \frac{z}{8}
\]

Solving for \( z \):

\[
z = 8 \times \tan(54^\circ)
\]

Let me calculate the value of \( z \).import math

# Given values
angle_degrees = 54
adjacent_side = 8

# Convert angle to radians for the math.tan function
angle_radians = math.radians(angle_degrees)

# Calculate the length of the opposite side (z)
z = adjacent_side * math.tan(angle_radians)
zThe value of \( z \) is approximately \( 11.0 \) cm, rounded to one decimal place.

Would you like more details on the steps, or do you have any further questions?

Here are some related questions to explore:
1. How can sine and cosine be used in this triangle to find other side lengths?
2. What are the different trigonometric ratios in a right triangle?
3. How would the calculation change if a different angle was given instead of 54°?
4. Can you use the Pythagorean theorem here, and if so, how?
5. What are the common mistakes when using trigonometric functions?

**Tip:** Always ensure that your calculator is in the correct mode (degrees or radians) when solving trigonometric problems.

Determine the value of z in the diagram where the angle is 54° and the adjacent side is 8 cm.

Learn how to determine the length of the opposite side in a right-angled triangle using the tangent function. The problem involves a 54° angle and an adjacent side of 8 cm, making it suitable for students in Grades 9-10.

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