๐ Topic 3.4 โ Sine & Cosine Graphs
Graph of f(ฮธ) = sin ฮธ
Input = angle ฮธ on horizontal axis ยท output = sin ฮธ on vertical axis
Building the sine graph from the unit circle
Since angle measures in standard position are periodic, f(ฮธ) = sin ฮธ is periodic too. We use ฮธ on the horizontal axis and sin ฮธ (the y-coordinate on the unit circle) on the vertical axis. Every angle from the unit circle gives us a point on the graph.
f(ฮธ) = sin ฮธ (red) and g(ฮธ) = cos ฮธ (blue dashed) โ one full period 0 to 2ฯ
| ฮธ | 0 | ฯ6 | ฯ3 | ฯ2 | 2ฯ3 | 5ฯ6 | ฯ | 7ฯ6 | 3ฯ2 | 5ฯ3 | 11ฯ6 | 2ฯ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| sin ฮธ | 0 | ยฝ | โ32 | 1 | โ32 | ยฝ | 0 | โยฝ | โ1 | โโ32 | โยฝ | 0 |
| Properties of f(ฮธ) = sin ฮธ | |||
|---|---|---|---|
| Midline Halfway between max and min |
Amplitude Distance from midline to max/min |
Period Length of one full cycle |
Frequency Reciprocal of the period |
| y = 0 | a = 1 | P = 2ฯ | 1/(2ฯ) |
Key shape facts for sin ฮธ
Starts at (0, 0) ยท rises to max (ฯ2, 1) ยท returns to zero at (ฯ, 0) ยท falls to min (3ฯ2, โ1) ยท returns to zero at (2ฯ, 0).
The graph oscillates between concave down (first half) and concave up (second half).
Graph of g(ฮธ) = cos ฮธ
Same shape as sin ฮธ, but shifted left by ฯ2 ยท starts at the maximum
| Properties of g(ฮธ) = cos ฮธ | |||
|---|---|---|---|
| Midline | Amplitude | Period | Frequency |
| y = 0 | a = 1 | P = 2ฯ | 1/(2ฯ) |
Key shape facts for cos ฮธ
Starts at maximum (0, 1) ยท crosses zero at (ฯ2, 0) ยท reaches min at (ฯ, โ1) ยท crosses zero at (3ฯ2, 0) ยท returns to max at (2ฯ, 1).
The graph also oscillates between concave down and concave up.