Graphs of trigonometry fuctions

The general form of a sine function is $y = A \sin(k\theta)$. Here, $A=1$, so the amplitude is 1.
The period $P$ is given by $P = \frac{360^\circ}{|k|}$. In this case, $k=3$.

This means one complete cycle of the graph occurs over an interval of $120^\circ$.

We find the values of $y$ at intervals of $\frac{P}{4} = \frac{120^\circ}{4} = 30^\circ$.

• At $\theta = 0^\circ$: $y = \sin(3 \cdot 0^\circ) = \sin(0^\circ) = 0$
• At $\theta = 30^\circ$: $y = \sin(3 \cdot 30^\circ) = \sin(90^\circ) = 1$ (maximum)
• At $\theta = 60^\circ$: $y = \sin(3 \cdot 60^\circ) = \sin(180^\circ) = 0$
• At $\theta = 90^\circ$: $y = \sin(3 \cdot 90^\circ) = \sin(270^\circ) = -1$ (minimum)
• At $\theta = 120^\circ$: $y = \sin(3 \cdot 120^\circ) = \sin(360^\circ) = 0$

Draw a coordinate plane with the horizontal axis representing $\theta$ (from $0^\circ$ to $120^\circ$) and the vertical axis representing $y$ (from -1 to 1). Plot the key points determined in Step 2. Connect these points with a smooth curve. The graph starts at $(0^\circ, 0)$, rises to a maximum of 1 at $30^\circ$, crosses the $\theta$-axis at $60^\circ$, falls to a minimum of -1 at $90^\circ$, and returns to the $\theta$-axis at $120^\circ$. This completes one cycle.

The general form of a cosine function is $y = A \cos(k\theta)$. Here, $A=1$, so the amplitude is 1.
The period $P$ is given by $P = \frac{360^\circ}{|k|}$. In this case, $k=3$.

This means one complete cycle of the graph occurs over an interval of $120^\circ$.

We find the values of $y$ at intervals of $\frac{P}{4} = \frac{120^\circ}{4} = 30^\circ$.

• At $\theta = 0^\circ$: $y = \cos(3 \cdot 0^\circ) = \cos(0^\circ) = 1$ (maximum)
• At $\theta = 30^\circ$: $y = \cos(3 \cdot 30^\circ) = \cos(90^\circ) = 0$
• At $\theta = 60^\circ$: $y = \cos(3 \cdot 60^\circ) = \cos(180^\circ) = -1$ (minimum)
• At $\theta = 90^\circ$: $y = \cos(3 \cdot 90^\circ) = \cos(270^\circ) = 0$
• At $\theta = 120^\circ$: $y = \cos(3 \cdot 120^\circ) = \cos(360^\circ) = 1$ (maximum)

Draw a coordinate plane with the horizontal axis representing $\theta$ (from $0^\circ$ to $120^\circ$) and the vertical axis representing $y$ (from -1 to 1). Plot the key points determined in Step 2. Connect these points with a smooth curve. The graph starts at $(0^\circ, 1)$, crosses the $\theta$-axis at $30^\circ$, falls to a minimum of -1 at $60^\circ$, crosses the $\theta$-axis again at $90^\circ$, and returns to a maximum of 1 at $120^\circ$. This completes one cycle.