Enter Graph Values
Interactive Graph
Generated Values Table
The table below shows the first plotted points from the selected range.
| x | Sine y | Cosine y |
|---|---|---|
| -6.28318531 | 0 | 1 |
| -6.17846555 | 0.10452846 | 0.9945219 |
| -6.0737458 | 0.20791169 | 0.9781476 |
| -5.96902604 | 0.30901699 | 0.95105652 |
| -5.86430629 | 0.40673664 | 0.91354546 |
| -5.75958653 | 0.5 | 0.8660254 |
| -5.65486678 | 0.58778525 | 0.80901699 |
| -5.55014702 | 0.66913061 | 0.74314483 |
| -5.44542727 | 0.74314483 | 0.66913061 |
| -5.34070751 | 0.80901699 | 0.58778525 |
| -5.23598776 | 0.8660254 | 0.5 |
| -5.131268 | 0.91354546 | 0.40673664 |
| -5.02654825 | 0.95105652 | 0.30901699 |
| -4.92182849 | 0.9781476 | 0.20791169 |
| -4.81710874 | 0.9945219 | 0.10452846 |
Example Data Table
This example uses A = 1, B = 1, h = 0, and D = 0 in radians.
| x | sin(x) | cos(x) | Meaning |
|---|---|---|---|
| 0 | 0 | 1 | Sine starts at midline. Cosine starts at peak. |
| π / 2 | 1 | 0 | Sine reaches peak. Cosine crosses midline. |
| π | 0 | -1 | Sine returns to midline. Cosine reaches trough. |
| 3π / 2 | -1 | 0 | Sine reaches trough. Cosine crosses midline. |
| 2π | 0 | 1 | One full cycle is complete. |
Formula Used
Sine form: y = A sin(B(x - h)) + D
Cosine form: y = A cos(B(x - h)) + D
Amplitude: |A|
Midline: y = D
Maximum: D + |A|
Minimum: D - |A|
Period in radians: 2π / |B|
Period in degrees: 360 / |B|
Frequency: 1 / period
How to Use This Calculator
- Select sine, cosine, or both curves.
- Choose radians or degrees for your x values.
- Enter amplitude, period coefficient, phase shift, and vertical shift.
- Set the x range and number of plotted sample points.
- Enter an x value for exact evaluation.
- Press the calculate button.
- Review the result cards, chart, and generated table.
- Use CSV or PDF export for saving your graph data.
Understanding Sine and Cosine Graphs
Wave Basics
Sine and cosine graphs describe smooth repeated motion. They appear in waves, circles, signals, tides, sound, and many school math problems. This calculator helps you see each part of the equation. It builds a table and a graph from the same values, so the picture and numbers match.
Core Shape
The standard sine wave starts at the midline. The standard cosine wave starts at a peak. Both waves repeat forever. Their normal amplitude is one. Their normal period is two pi radians, or three hundred sixty degrees. Changing the equation stretches, shifts, or moves the wave.
Amplitude and Midline
Amplitude controls height above and below the midline. A larger amplitude makes taller peaks and lower troughs. A negative amplitude reflects the graph. The vertical shift sets the midline. When the shift changes, the whole wave moves up or down without changing its width.
Period and Phase
The coefficient before x controls the period. A larger coefficient makes the wave repeat faster. A smaller coefficient makes it repeat slower. Phase shift moves the wave left or right. This is useful when two waves have the same shape but start at different times.
Why Tables Matter
A graph is visual, but a table gives exact checkpoints. The sampled points show how x values become y values. You can export these points and use them in reports, homework, spreadsheets, or experiments. More samples make the curve smoother, but very large tables can be harder to read.
Using Both Curves
Plotting sine and cosine together helps compare timing. Cosine leads sine by a quarter cycle in the standard position. With custom phase values, you can study delay, lead, and interference. This is useful for physics, electronics, trigonometry, and signal modeling.
Reading Results
Look at amplitude, period, frequency, midline, and selected point values first. Then inspect the graph. Peaks mark maximum values. Troughs mark minimum values. Zero crossings show where the wave meets the horizontal axis. Use the CSV and PDF buttons to save your work.
For clearer study, test one change at a time. Small changes reveal how each parameter shapes the curve and affects every sampled point shown.
FAQs
1. What does amplitude mean?
Amplitude is the distance from the midline to a peak or trough. In the equation, it is the absolute value of A. Larger amplitude creates a taller wave.
2. What does the B value control?
The B value controls the period. In radians, the period is 2π divided by |B|. In degrees, the period is 360 divided by |B|.
3. What is phase shift?
Phase shift moves the graph left or right. In this calculator, h is subtracted from x. A positive h usually shifts the curve to the right.
4. What is vertical shift?
Vertical shift moves the entire graph up or down. It also defines the midline. If D equals 3, the wave oscillates around y = 3.
5. Should I use degrees or radians?
Use degrees for common angle measures like 90 or 360. Use radians for calculus, advanced trigonometry, and many physics problems.
6. Why do sine and cosine look shifted?
Sine and cosine have the same shape, but different starting points. Standard cosine starts at a peak. Standard sine starts at the midline.
7. Why increase sample points?
More sample points make the plotted curve smoother. Fewer points are faster and easier to read. A balanced value works best for most graphs.
8. What does the export option save?
The CSV export saves plotted x, sine, and cosine values. The PDF export saves the summary, chart, and key graph data for later review.