The Mother Curve
Where does the shape of sin x come from? Built from the unit circle, not memorised. The five key points that anchor every trig graph.
Where does the shape of sin x come from?
Not from a textbook shape to memorise. From a point moving around a circle.
The unit circle argument
Place a point P on a circle of radius 1, centred at the origin. Let θ be the angle P makes with the positive x-axis. Then sin θ is simply the height of P above the x-axis.
Now let θ increase from 0 to 2π — one full revolution. Track the height:
| θ | Position of P | sin θ |
|---|---|---|
| 0 | Rightmost point | 0 |
| π/2 | Top | 1 |
| π | Leftmost point | 0 |
| 3π/2 | Bottom | −1 |
| 2π | Back to start | 0 |
These five points — (0, 0), (π/2, 1), (π, 0), (3π/2, −1), (2π, 0) — are the skeleton of every sin graph. Everything else, including all the transformations ahead, hangs on these five points.
Connect them with a smooth curve. That is y = sin x.
What the curve tells you
- It oscillates between −1 and +1. Never above, never below.
- It repeats every 2π. This is the period.
- It passes through the origin going upward. This distinguishes sin from cos.
sin x vs cos x
cos x is sin x shifted left by π/2. That is the only difference:
Students who understand this never treat sin and cos as separate shapes to memorise separately.
Five key points to plot any sin or cos graph
Before touching any transformation, practice drawing y = sin x from scratch using only the five key points. Under exam pressure, this is your anchor.
- Start at (0, 0) — crosses the x-axis going up
- Peak at (π/2, 1)
- Back to zero at (π, 0) — crosses going down
- Trough at (3π/2, −1)
- Back to zero at (2π, 0)
Draw the curve through these five points. That is the mother curve. Everything in the rest of this series is a transformation of this one shape.
Next: The four-parameter machine — what A, B, C, D each do to this curve.