Calculator Input
Formula Used
The calculator uses the transformed sine model:
y = A sin(Bx + C) + D
| Property | Formula |
|---|---|
| Amplitude | |A| |
| Period in radians | 2π / |B| |
| Period in degrees | 360 / |B| |
| Phase shift | -C / B |
| Vertical shift | D |
| Range | [D - |A|, D + |A|] |
| Frequency | 1 / Period |
How to Use This Calculator
- Enter the A coefficient to control amplitude and reflection.
- Enter the B coefficient to control period and horizontal compression.
- Enter the C coefficient to calculate phase movement.
- Enter the D coefficient to shift the curve vertically.
- Select degrees or radians before entering C and x values.
- Set an interval and sample count for the data table.
- Press calculate to show results below the header.
- Use CSV or PDF export for records and class work.
About Sine Function Properties
Why Sine Properties Matter
A sine function describes repeated motion. It appears in waves, sound, voltage, tides, circular motion, and many geometry problems. The basic curve is simple. Transformations make it more useful. A coefficient can stretch it. Another can compress it. A constant can move it left, right, up, or down.
Main Parts of the Model
This calculator studies the form y = A sin(Bx + C) + D. The value A controls amplitude. It shows the distance from the midline to a peak. A negative A also reflects the curve. The value B changes the period. Large absolute B values make cycles shorter. Small absolute B values make cycles wider. The value C creates a phase shift. The value D creates the midline and vertical shift.
Interpreting the Results
The range is found from the amplitude and vertical shift. The maximum value is D plus the amplitude. The minimum value is D minus the amplitude. The period tells how long one full cycle takes. Frequency is the reciprocal of the period. These values help compare sine curves without drawing every point.
Using Degrees and Radians
Angle units change the period formula. In degrees, one basic cycle is 360 units. In radians, one basic cycle is 2π units. Keep your C value and x value in the same unit mode. Mixed units produce misleading answers. This is important in trigonometry, physics, and engineering tasks.
Advanced Practical Use
The sample table helps you inspect numerical behavior across an interval. The key point table gives standard positions for one transformed cycle. You can use these points to sketch the curve. You can also export results for reports, worksheets, or checking homework. The graph gives a quick visual check. It is helpful when a sign error or unit mistake changes the curve shape.
Example Data Table
| A | B | C | D | Unit | Amplitude | Period | Phase Shift | Range |
|---|---|---|---|---|---|---|---|---|
| 2 | 1 | 30 | 1 | Degrees | 2 | 360 | -30 | [-1, 3] |
| 3 | 2 | 0 | -1 | Degrees | 3 | 180 | 0 | [-4, 2] |
| -4 | 0.5 | 45 | 2 | Degrees | 4 | 720 | -90 | [-2, 6] |
FAQs
1. What does amplitude mean?
Amplitude is the distance from the midline to a maximum or minimum point. For y = A sin(Bx + C) + D, it equals |A|.
2. How is the period calculated?
The period is 2π / |B| in radians. In degrees, the period is 360 / |B|. It measures one complete cycle.
3. What is phase shift?
Phase shift is the horizontal movement of the sine curve. For this form, it equals -C / B.
4. What does D do in the function?
D moves the whole curve up or down. It also sets the midline of the sine wave.
5. Why does a negative A matter?
A negative A reflects the sine curve across its midline. The amplitude stays positive because amplitude uses |A|.
6. Can I use radians?
Yes. Select radians before entering C, x, and interval values. Keep all angle-based inputs in the same unit system.
7. What does frequency mean here?
Frequency is the reciprocal of the period. It tells how many cycles occur per one x unit.
8. Why is B not allowed to be zero?
When B is zero, the expression has no normal sine cycle. Period and phase shift are not defined in the usual way.