Calculator Input
Formula Used
Sine form: y = A sin(B(x - C)) + D
Cosine form: y = A cos(B(x - C)) + D
Tangent form: y = A tan(B(x - C)) + D
A is amplitude for sine and cosine. It is vertical stretch for tangent. A negative value reflects the graph over its midline.
B controls period. For sine and cosine, B = 2π / period in radians or B = 360 / period in degrees.
For tangent, B = π / period in radians or B = 180 / period in degrees. C is the phase shift. D is the vertical shift.
How to Use This Calculator
- Select sine, cosine, or tangent as the required function family.
- Choose direct feature mode or extrema mode.
- Enter amplitude, period, phase shift, and vertical shift.
- Use maximum and minimum values when the graph gives extrema.
- Pick radians or degrees for the x-axis scale.
- Set the graph domain and decimal precision.
- Press the submit button to generate the equation.
- Download the result as CSV or PDF for records.
Example Data Table
| Case | Family | Amplitude / Stretch | Period | Phase Shift | Vertical Shift | Result Form |
|---|---|---|---|---|---|---|
| Basic wave | Sine | 3 | 2π | 0 | 1 | y = 3sin(x) + 1 |
| Shifted wave | Cosine | 4 | 8 | 2 | -1 | y = 4cos(0.7854(x - 2)) - 1 |
| Reflected model | Sine | -2 | 6 | -1 | 0 | y = -2sin(1.0472(x + 1)) |
| Tangent model | Tangent | 1.5 | π | 0 | 2 | y = 1.5tan(x) + 2 |
Understanding Trig Functions From Features
Why Features Matter
A trigonometric graph can look complex at first. Yet most of its shape comes from four simple features. These are amplitude, period, phase shift, and vertical shift. When these values are known, the equation can be written with confidence. This calculator turns those features into a clean sine, cosine, or tangent model. It also shows key points, range, domain notes, and a visual graph.
Amplitude and Vertical Shift
For sine and cosine, amplitude measures the distance from the midline to a peak. If a graph has a maximum and minimum, amplitude is half of their difference. The vertical shift is the midline. It is found by averaging the maximum and minimum. Tangent has no maximum or minimum, so its multiplier is treated as vertical stretch.
Period and Horizontal Scale
Period tells how far the graph travels before repeating. Sine and cosine complete a full cycle after one period. Tangent repeats between matching branches. The coefficient B changes the horizontal scale. A larger B compresses the graph. A smaller B stretches it. This tool supports radians and degrees, so classroom and engineering styles both work.
Phase Shift and Reflection
The phase shift moves the graph left or right. In the form x minus C, a positive C shifts the curve right. A negative C shifts it left. Reflection changes the sign of A. For sine and cosine, this flips peaks and troughs. For tangent, it reverses the branch direction around the center point.
Using the Output
The final equation is useful for homework, lesson planning, graph checks, and model building. The key points table helps verify the result manually. The graph gives an immediate visual check. CSV export is useful for spreadsheets. PDF export is useful for sharing or saving a clean report. Always compare the generated curve with the original graph features before using it in final work.
FAQs
1. What does this calculator do?
It writes a sine, cosine, or tangent function from graph features like amplitude, period, phase shift, and vertical shift.
2. Can I use maximum and minimum values?
Yes. Select extrema mode for sine or cosine. The tool finds amplitude and midline from the maximum and minimum values.
3. Why is tangent different?
Tangent has no maximum or minimum. It uses vertical stretch, period, center shift, and vertical shift instead.
4. What is phase shift?
Phase shift is the horizontal movement of the graph. Positive values move the model right in the displayed equation.
5. What does B mean?
B controls the horizontal scale. It is calculated from the period using formulas for sine, cosine, or tangent.
6. Can I work in degrees?
Yes. Choose degrees as the angle unit. The calculator then uses degree-based period formulas for the coefficient B.
7. What does reflection do?
Reflection makes the multiplier negative. This flips sine and cosine over the midline and reverses tangent branch direction.
8. Why download CSV or PDF?
CSV helps with spreadsheet checks. PDF gives a clean report for homework, teaching notes, or saved project records.