numpy as np import matplotlib.pyplot as plt

Define the interval and functions

x = np.linspace(-2 * np.pi, 2 * np.pi, 1000) y1 = np.sin(x) y2 = 1/2

Plot the functions

plt.figure(figsize=(10, 6)) plt.plot(x, y1, label='y1 = sin(x)') plt.axhline(y=y2, color='r', linestyle='--', label='y2 = 1/2')

Highlight intersection points

intersection_x = x[np.isclose(y1, y2, atol=0.01)] intersection_y = y1[np.isclose(y1, y2, atol=0.01)]

plt.scatter(intersection_x, intersection_y, color='purple', label='Intersection points')

Add labels and legend

plt.title('Graph of y1 = sin(x) and y2 = 1/2') plt.xlabel('x') plt.ylabel('y') plt.axhline(0, color='black', linewidth=0.8) plt.axvline(0, color='black', linewidth=0.8) plt.grid(alpha=0.3) plt.legend()

Display the plot

plt.show()

Return intersection points

intersection_xThe intersections occur at the following x-values (approximately):

• $x = -5.77, -5.75, -3.67, -3.65, 0.52, 0.53, 2.61, 2.62$.

These are the solutions where $y_1 = \sin(x) = y_2 = \frac{1}{2}$ in the interval $[-2\pi, 2\pi]$.

Would you like more details or have any specific questions?

Here are 5 related questions to deepen your understanding:

1. How can we find the exact solutions to this equation using trigonometric identities?
2. What is the periodicity of the sine function, and how does it help determine these solutions?
3. Why is the line $y_2 = \frac{1}{2}$ horizontal?
4. How would the solutions change if $y_2$ were a different value?
5. Can you explain the difference between numerical and analytical solutions?

Tip: Always consider the periodic nature of trigonometric functions when solving equations over specific intervals!