Sure, let's walk through how you would plot the points for the sine function [tex]\( y = \sin(\theta) \)[/tex] and the cosine function [tex]\( y = \cos(\theta) \)[/tex], where [tex]\(\theta\)[/tex] is the central angle, using values from the unit circle.
#### Step-by-Step Process
1. Unit Circle Basics:
- The unit circle is a circle with a radius of 1 centered at the origin (0, 0) of a coordinate plane.
- Any point [tex]\((x, y)\)[/tex] on the unit circle can be described in terms of the central angle [tex]\(\theta\)[/tex] as [tex]\(\theta\)[/tex] varies from [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex] radians.
- The x-coordinate of each point on the unit circle is the cosine of [tex]\(\theta\)[/tex], and the y-coordinate is the sine of [tex]\(\theta\)[/tex].
2. Generate Points with [tex]\(\theta\)[/tex] Angles:
- We consider [tex]\( \theta \)[/tex] values between [tex]\( 0 \)[/tex] and [tex]\( 2\pi \)[/tex] radians.
- We divide this interval into smaller steps to get a smoother graph. For example, using [tex]\(100\)[/tex] equal intervals, we get:
[tex]\[
\theta = 0, 0.0635, 0.127, 0.1904, \ldots, 6.2197, 6.2832
\][/tex]
3. Calculate Sine and Cosine Values:
- For each [tex]\(\theta\)[/tex] value computed, we find [tex]\(\sin(\theta)\)[/tex] and [tex]\(\cos(\theta)\)[/tex].
- This results will be pairs of [tex]\((\theta, \sin(\theta))\)[/tex] and [tex]\((\theta, \cos(\theta))\)[/tex].
4. Example Sine Values:
- When [tex]\(\theta = 0\)[/tex], [tex]\(\sin(0) = 0\)[/tex]
- When [tex]\(\theta = 0.0635\)[/tex], [tex]\(\sin(0.0635) \approx 0.0634\)[/tex]
- When [tex]\(\theta = 0.127\)[/tex], [tex]\(\sin(0.127) \approx 0.1269\)[/tex]
- ...
- When [tex]\(\theta = 6.2832\)[/tex], [tex]\(\sin(6.2832) \approx 0\)[/tex]
Here is a small sample of calculated sine values:
[tex]\[
\begin{array}{|c|c|}
\hline
\theta & \sin(\theta) \\
\hline
0 & 0 \\
0.0635 & 0.0634 \\
0.127 & 0.1269 \\
\cdots & \cdots \\
6.2832 & 0 \\
\hline
\end{array}
\][/tex]
5. Example Cosine Values:
- When [tex]\(\theta = 0\)[/tex], [tex]\(\cos(0) = 1\)[/tex]
- When [tex]\(\theta = 0.0635\)[/tex], [tex]\(\cos(0.0635) \approx 0.998\)[/tex]
- When [tex]\(\theta = 0.127\)[/tex], [tex]\(\cos(0.127) \approx 0.992\)[/tex]
- ...
- When [tex]\(\theta = 6.2832\)[/tex], [tex]\(\cos(6.2832) \approx 1\)[/tex]
Here is a small sample of calculated cosine values:
[tex]\[
\begin{array}{|c|c|}
\hline
\theta & \cos(\theta) \\
\hline
0 & 1 \\
0.0635 & 0.9980 \\
0.127 & 0.9920 \\
\cdots & \cdots \\
6.2832 & 1 \\
\hline
\end{array}
\][/tex]
6. Plotting the Points:
- Using the computed values, we now plot the points [tex]\((\theta, \sin(\theta))\)[/tex] and [tex]\((\theta, \cos(\theta))\)[/tex] on the coordinate plane.
- Connect these points smoothly to form the complete graphs of [tex]\( y = \sin(\theta) \)[/tex] and [tex]\( y = \cos(\theta) \)[/tex].
#### Graphing the Functions
Now, visualize these points and imagine drawing a smooth, continuous curve through them. The sine curve starts at [tex]\((0, 0)\)[/tex], reaches its maximum value [tex]\(1\)[/tex] at [tex]\(\pi/2\)[/tex] radians, and its minimum value [tex]\(-1\)[/tex] at [tex]\(3\pi/2\)[/tex] radians, and returns to [tex]\(0\)[/tex] at [tex]\(2\pi\)[/tex] radians. The cosine curve starts at [tex]\((0, 1)\)[/tex], reaches [tex]\(0\)[/tex] at [tex]\(\pi/2\)[/tex] radians, the minimum at [tex]\(\pi\)[/tex] radians, [tex]\(0\)[/tex] again at [tex]\(3\pi/2\)[/tex], and back to [tex]\(1\)[/tex] at [tex]\(2\pi\)[/tex].
This process showcases how the unit circle forms the basis of trigonometric functions and helps us visualize the periodic nature of sine and cosine graphs.
