1. **Problem Statement:** We are asked to sketch and analyze the graphs of sine and cosine functions with horizontal shifts (phase shifts), amplitude changes, and combined transformations. 2. **Horizontal Shifts (Phase Shift 'C')** The general form for a phase shift is: $$y = \sin(x - C) \quad \text{or} \quad y = \cos(x + C)$$ where $C$ is the phase shift in degrees. - A positive $C$ in $\sin(x - C)$ shifts the graph to the right by $C$ units. - A positive $C$ in $\cos(x + C)$ shifts the graph to the left by $C$ units. 3. **Amplitude Changes ('A')** The general form for amplitude change is: $$y = A \sin(x) \quad \text{or} \quad y = A \cos(x)$$ where $|A|$ is the amplitude. - If $A > 0$, the graph stretches vertically by a factor of $A$. - If $A < 0$, the graph reflects about the x-axis and stretches by $|A|$. 4. **Exact Values of Sine and Cosine** From the given table: $$\sin(0^\circ) = 0, \sin(30^\circ) = \frac{1}{2}, \sin(45^\circ) = \frac{\sqrt{2}}{2}, \sin(60^\circ) = \frac{\sqrt{3}}{2}, \sin(90^\circ) = 1$$ $$\cos(0^\circ) = 1, \cos(30^\circ) = \frac{\sqrt{3}}{2}, \cos(45^\circ) = \frac{\sqrt{2}}{2}, \cos(60^\circ) = \frac{1}{2}, \cos(90^\circ) = 0$$ 5. **Phase Shift Application Example:** Choose $K = 60^\circ$. - For $y = \sin(x - 60^\circ)$, values are computed by substituting $x - 60^\circ$ into sine values. - For $y = \cos(x + 60^\circ)$, values are computed by substituting $x + 60^\circ$ into cosine values. 6. **General Rules:** - $y = \sin(x - C)$ shifts the sine graph right by $C$ if $C > 0$, left if $C < 0$. - $y = \cos(x + C)$ shifts the cosine graph left by $C$ if $C > 0$, right if $C < 0$. - $y = A \sin(x)$ and $y = A \cos(x)$ have amplitude $|A|$; negative $A$ reflects the graph vertically. 7. **Final Answer:** The transformations of sine and cosine functions involve horizontal shifts by $C$ units and vertical stretches/reflections by $A$. These changes model ocean wave behaviors such as starting point shifts and wave height changes.