Calculator Input
Example Data Table
This example uses y = cos²(x) in degrees.
| x | cos(x) | cos²(x) | Meaning |
|---|---|---|---|
| 0° | 1 | 1 | Maximum value |
| 30° | 0.866025 | 0.75 | High positive square |
| 45° | 0.707107 | 0.5 | Midpoint value |
| 60° | 0.5 | 0.25 | Lower value |
| 90° | 0 | 0 | Minimum value |
| 180° | -1 | 1 | Maximum repeats |
Formula Used
The calculator uses this transformed cosine squared model:
y = A cos²(Bx + C) + D
The value inside the cosine is first calculated as Bx + C.
Then the cosine is found. That cosine value is squared.
The result is multiplied by A and shifted by D.
The identity form is also useful:
cos²(θ) = (1 + cos(2θ)) / 2
This identity shows why the period is shorter than regular cosine.
The base period of cos²(x) is π radians or 180°.
With multiplier B, the period becomes π / |B| or 180° / |B|.
How to Use This Calculator
- Enter the amplitude multiplier A.
- Enter the frequency multiplier B.
- Add a phase constant C when needed.
- Add a vertical shift D when needed.
- Select degrees or radians.
- Enter the x-value that you want to evaluate.
- Set the graph start, graph end, and step size.
- Press Calculate and Graph.
- Review the result, graph, and generated table.
- Use CSV or PDF download for saving results.
Understanding Cos Squared Graphs
What the Curve Shows
A cos squared graph shows the square of a cosine value. This makes every output nonnegative before transformations. Regular cosine moves between negative and positive values. Cos squared removes the sign by squaring the value. So the basic curve stays between zero and one. It touches one when cosine equals one or negative one. It touches zero when cosine equals zero.
Why the Period Changes
The basic cosine graph has a full period of 360 degrees. In radians, that period is two pi. Cos squared repeats faster than regular cosine. Its base period is 180 degrees, or pi radians. This happens because negative cosine values become positive after squaring. The second half of a cosine wave mirrors the first half. The graph therefore repeats in half the usual distance.
Advanced Transformations
The model uses four major controls. The value A changes vertical size. If A is negative, the curve reflects downward. The value B compresses or stretches the graph horizontally. Larger B values produce shorter periods. Smaller B values produce wider cycles. The value C creates a phase shift. It moves the curve left or right. The value D shifts the graph vertically. It changes the minimum and maximum levels.
Using Tables with Graphs
A table helps confirm the curve. Each row gives one x-value and one output. Smaller step sizes make smoother graphs. Larger step sizes make shorter tables. For study work, a balanced step is best. Use common angles like 0, 30, 45, 60, and 90 degrees. These values reveal the main shape quickly.
Practical Math Uses
Cos squared functions appear in trigonometry and wave analysis. They are also used in physics, signal study, and oscillation models. The curve is useful when direction does not matter. Squaring measures size without keeping the sign. This calculator helps compare equations, check homework, and export data. It also supports quick graph inspection. Use the PDF and CSV buttons when you need records.
FAQs
1. What does cos squared mean?
Cos squared means the cosine value is multiplied by itself. It is written as cos²(x), which equals cos(x) × cos(x).
2. Is cos²(x) the same as cos(x²)?
No. cos²(x) means the cosine result is squared. cos(x²) means x is squared first, then cosine is applied.
3. What is the period of cos²(x)?
The base period is 180 degrees or π radians. With multiplier B, the period becomes 180° / |B| or π / |B|.
4. Why is the basic cos squared graph never negative?
Squaring removes negative signs. Since cosine is squared, the basic output stays from 0 to 1 before transformations.
5. What does A do in the equation?
A changes the vertical size of the graph. A negative value also reflects the curve across its shifted center behavior.
6. What does B do in the equation?
B controls horizontal compression or stretching. Larger absolute B values create shorter periods and faster repeating cycles.
7. Can I use radians instead of degrees?
Yes. Select radians in the unit field. Then x-values and the phase constant are treated as radian measurements.
8. Why use CSV and PDF exports?
CSV is useful for spreadsheets and data analysis. PDF is useful for reports, assignments, notes, and saved graph summaries.