This lesson introduces the fundamental trigonometric graphs, $y = \sin(x)$ and $y = \cos(x)$, by showing how they are derived from the unit circle and exploring their key features.
The Unit Circle
The unit circle is a circle with a radius of 1 centered at the origin $(0,0)$ of a coordinate plane. For any point $(x, y)$ on the unit circle that corresponds to an angle $\theta$ (measured counter-clockwise from the positive x-axis), the coordinates are defined as:
- $x = \cos(\theta)$
- $y = \sin(\theta)$
This relationship is the foundation for graphing trigonometric functions. As the angle $\theta$ changes, the values of $\sin(\theta)$ and $\cos(\theta)$ oscillate between -1 and 1.
Graphing the Parent Function y = sin(x)
To graph $y = \sin(x)$, we plot the angle (represented by $x$) on the horizontal axis and the sine value (the y-coordinate from the unit circle) on the vertical axis. Let's look at key points for one full rotation around the circle ($0$ to $2\pi$):
- When $x = 0$, $\sin(0) = 0$.
- When $x = \frac{\pi}{2}$ (90°), $\sin(\frac{\pi}{2}) = 1$.
- When $x = \pi$ (180°), $\sin(\pi) = 0$.
- When $x = \frac{3\pi}{2}$ (270°), $\sin(\frac{3\pi}{2}) = -1$.
- When $x = 2\pi$ (360°), $\sin(2\pi) = 0$.
Plotting these points and connecting them with a smooth curve gives us the characteristic 'wave' shape of the sine function.
Graphing the Parent Function y = cos(x)
Similarly, to graph $y = \cos(x)$, we plot the angle on the horizontal axis and the cosine value (the x-coordinate from the unit circle) on the vertical axis.
- When $x = 0$, $\cos(0) = 1$.
- When $x = \frac{\pi}{2}$ (90°), $\cos(\frac{\pi}{2}) = 0$.
- When $x = \pi$ (180°), $\cos(\pi) = -1$.
- When $x = \frac{3\pi}{2}$ (270°), $\cos(\frac{3\pi}{2}) = 0$.
- When $x = 2\pi$ (360°), $\cos(2\pi) = 1$.
The cosine graph has the same wave shape as the sine graph, but it is shifted horizontally.
Key Features of Sine and Cosine Graphs
- Domain: The domain is the set of all possible input values (x-values). For both $y = \sin(x)$ and $y = \cos(x)$, you can use any real number as the angle. Therefore, the domain is All Real Numbers, or $(-\infty, \infty)$.
- Range: The range is the set of all possible output values (y-values). Because the sine and cosine values come from the coordinates on the unit circle (with radius 1), their values can never be greater than 1 or less than -1. The range for both functions is $[-1, 1]$.
- Zeros: Zeros are the points where the graph crosses the x-axis (where $y=0$).
- For $y = \sin(x)$, the zeros occur at integer multiples of $\pi$. We can write this as $x = k\pi$, where $k$ is any integer (e.g., $...-2\pi, -\pi, 0, \pi, 2\pi...$).
- For $y = \cos(x)$, the zeros occur at odd multiples of $\frac{\pi}{2}$. We can write this as $x = \frac{\pi}{2} + k\pi$, where $k$ is any integer (e.g., $...-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}...$).
- Periodicity: A function is periodic if its graph repeats at regular intervals. The length of one complete cycle is called the period. For both $y = \sin(x)$ and $y = \cos(x)$, the graph completes one full cycle every $2\pi$ radians. So, the period is $2\pi$.
- Amplitude: The amplitude is the distance from the center line of the wave to either its maximum or minimum point. It measures the 'height' of the wave. It can be calculated as half the distance between the maximum and minimum values.
- Amplitude = $\frac{\text{Maximum Value} - \text{Minimum Value}}{2}$
- For both parent functions, Amplitude = $\frac{1 - (-1)}{2} = \frac{2}{2} = 1$.
