The constellation points and the phases you see in this solution likely come from a form of quadrature amplitude modulation (QAM) or phase-shift keying (PSK). In these systems, different symbols (or signals) are represented by different points on a constellation diagram, which maps them to specific phase angles and amplitudes.

Why 225°?

In phase modulation schemes, the angle (or phase) of the signal is one of the main characteristics that differentiate symbols. In this case, the 225° phase corresponds to a specific point on the constellation diagram. Here's why:

1. Sine and Cosine Components: The expression for $y_1(t)$ includes a combination of sine and cosine functions, which describe the phase of the signal: $y_1(t) = -\cos(2\pi f_c t) - \sin(2\pi f_c t)$ This combination of sine and cosine corresponds to a point in the third quadrant of the unit circle, which is around 225°.

2. Quadrants and Phase Angles:

   • The cosine function represents the in-phase component (x-axis on the constellation diagram).
   • The sine function represents the quadrature component (y-axis on the constellation diagram).
   • The negative signs in both the cosine and sine functions push the vector into the third quadrant (between 180° and 270°).
   • In the third quadrant, the vector at an angle of 225° means that both the in-phase and quadrature components are negative.

3. Magnitude and Phase: The vector represented by $-\cos(2\pi f_c t) - \sin(2\pi f_c t)$ has a magnitude of $\sqrt{2}$, which confirms its distance from the origin (length of the vector), and the phase angle is 225° based on how the sine and cosine functions are combined.

Thus, 225° is the phase angle for this particular combination of sine and cosine terms.

Would you like further details or have any other questions about the constellation diagram?

Relative Questions:

1. How does the constellation diagram relate to QAM and PSK?
2. What is the significance of the magnitude $\sqrt{2}$ in this context?
3. How do different phase angles correspond to different signals in modulation schemes?
4. What other phase angles appear in this constellation diagram, and how are they derived?
5. How do in-phase and quadrature components define the signal in these systems?

Tip: Always consider the quadrant when determining phase angles from sine and cosine functions—each quadrant has distinct sign conventions for these functions.