Sine Curve Morphs into Cosine – Phase Shift Demo

Animation Specification for SinToCos

1. Overview

• Purpose: Visualize the relationship between the sine and cosine functions by first drawing the sine curve on a set of axes, then smoothly transforming it into the cosine curve.
• Target Audience: Students learning trigonometric functions and their phase shift relationship.

2. Mathematical Elements

• Axes:
  • x‑range: $[0, 2\pi]$
  • y‑range: $[-1.5, 1.5]$
  • x‑ticks: at multiples of $\frac{\pi}{2}$ (i.e., $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$) with labels using LaTeX (e.g., "$0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, $2\pi$").
  • y‑ticks: at $-1, 0, 1$ with labels "$-1$", "$0$", "$1$".
• Functions:
  • Sine: $y = \sin(x)$ plotted in BLUE.
  • Cosine: $y = \cos(x)$ plotted in RED.

3. Visual Elements

Element	Description	Color / Style
Axes	Standard Cartesian axes with arrowheads, tick marks, and labels as described above.	White (default)
Sine curve	Smooth curve representing $\sin(x)$.	BLUE (hex #1B9AF7)
Cosine curve	Smooth curve representing $\cos(x)$.	RED (hex #FF4C4C)
Background	Black or dark gray to make the colored curves stand out.	#0B0B0B (default Manim background)

4. Animation Sequence & Timing

Step	Action	Duration	Easing / Transition
1	Create Axes – draw the axes with a Create animation.	2 seconds	Linear (default)
2	Create Sine Curve – draw the sine curve using Create.	2 seconds	Linear
3	Transform to Cosine – morph the sine curve into the cosine curve using Transform.	3 seconds	Smooth (default)
4	Pause – hold the final frame so viewers can observe the cosine curve.	1.5 seconds	–

5. Camera Settings

• Default 2‑D camera (no zoom or rotation). The camera frame should be sized to comfortably include the entire axes range with a small margin (e.g., camera.frame_width = 8, camera.frame_height = 6).
• No additional camera movements are required.

6. Additional Details & Assumptions

• Line thickness for both curves: stroke_width=4 (default Manim line thickness).
• Axes labels: optional x and y labels can be added ("x" and "y") positioned near the ends of the axes.
• Background opacity: keep the default opaque background for clarity.
• Code Structure: The animation follows the provided class structure, with the construct method containing the sequence of self.play calls.
• Libraries: Assumes manim version 0.17+ and numpy imported as np.

7. Full Pseudocode (for reference)

class SinToCos(Scene):
    def construct(self):
        axes = Axes(
            x_range=[0, 2*PI, PI/2],
            y_range=[-1.5, 1.5, 0.5],
            x_axis_config={"numbers_to_include": [0, PI/2, PI, 3*PI/2, 2*PI]},
            y_axis_config={"numbers_to_include": [-1, 0, 1]},
            tips=False,
        )
        sin_curve = axes.plot(lambda x: np.sin(x), color=BLUE, stroke_width=4)
        cos_curve = axes.plot(lambda x: np.cos(x), color=RED, stroke_width=4)

        self.play(Create(axes), run_time=2)
        self.play(Create(sin_curve), run_time=2)
        self.play(Transform(sin_curve, cos_curve), run_time=3)
        self.wait(1.5)

This specification provides all necessary details for recreating the animation, including mathematical context, visual styling, timing, and camera setup.