To determine the value of $z$ in the given right triangle:

1. Given Information:

   • Angle: $52^\circ$
   • Adjacent side (to $52^\circ$): 12 cm
   • Opposite side (to $52^\circ$): $z$

2. Trigonometric Relation: Using the tangent function: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ Substitute the known values: $\tan(52^\circ) = \frac{z}{12}$

3. Solve for $z$: $z = 12 \cdot \tan(52^\circ)$

4. Calculation: Using a calculator: $\tan(52^\circ) \approx 1.2799$ $z = 12 \cdot 1.2799 \approx 15.4 \, \text{cm}$

Final Answer: $z = 15.4 \, \text{cm}$

Let me know if you'd like further clarification or more details! Here are five related questions to explore further:

1. How is the tangent function derived in trigonometry?
2. What are the other trigonometric functions used in solving triangles?
3. How can we calculate the hypotenuse of this triangle?
4. How would the value of $z$ change if the angle were smaller or larger?
5. What is the relationship between the angles and sides of a triangle in general?

Tip: Always check if your calculator is set to degrees when solving trigonometric problems with