Overview
This lecture covers the basics of the sine (sin) trigonometric graph, including how to draw it, key characteristics (period, amplitude, domain, range), and how these elements appear on the graph.
Drawing the Sine Graph
- The basic sine graph follows the equation y = sin x, with x measured in degrees.
- Plot key points at x = 0°, 90°, 180°, 270°, and 360°; y-values are 0, 1, 0, -1, and 0, respectively.
- Connect plotted points with smooth, curved lines, not straight segments.
- The graph is continuous and repeats its wave pattern indefinitely in both directions.
Key Features of the Sine Graph
- Turning points are labeled at local maximum and minimum points (e.g., (90°, 1), (270°, -1)).
- The graph extends infinitely, but the domain can be restricted (e.g., from 0° to 360°) depending on the question.
- The sine graph is periodic, creating repeating cycles.
Domain and Range
- Domain is the set of possible x-values for which the graph exists (e.g., [0°, 360°] or [-360°, 360°]).
- Range is the set of possible y-values; for the basic graph, it's [-1, 1], including these endpoints.
- Use square brackets [ ] to show values are included in the domain or range.
Period and Amplitude
- Period is the length of one full cycle; for sin x, this is always 360° in grade 10.
- Amplitude is the distance from the equilibrium (middle position) to the highest point; for y = sin x, amplitude is 1.
- Amplitude is calculated as (maximum y - minimum y) / 2.
- If the amplitude is negative, the graph reflects over the x-axis.
Changing the Graph
- A coefficient in front of sin (e.g., y = 2 sin x) stretches or compresses the graph vertically (changes amplitude).
- The period for grade 10 sine graphs remains 360°, regardless of amplitude changes.
Key Terms & Definitions
- Sine (sin) graph — The graph of y = sin x, showing a smooth wave that repeats every 360°.
- Domain — Set of x-values where the graph exists.
- Range — Set of y-values reached by the graph.
- Period — The horizontal length for one complete wave (360° for sin x).
- Amplitude — The maximum distance from the equilibrium to the peak (or trough) of the graph.
- Turning point — A maximum or minimum point on the graph.
Action Items / Next Steps
- Practice sketching the basic sine graph over different domains.
- Label turning points and calculate amplitude and period for given sine equations.
- Prepare for the next lesson on the cosine (cos) graph.