Calculator Inputs
Interactive Graph
The graph compares sin(θ), sin(2θ), and sin²(θ). Use it to inspect amplitude, period, intersections, and repeated behaviour. The display updates from the selected table range.
Formula Used
The double-angle formula combines sine and cosine values together. Sine squared instead multiplies sine by itself once. These expressions must never be treated as identical quantities.
How to Use
- Select the required calculation mode.
- Enter one angle or several separated values.
- Choose degrees, radians, or gradians.
- Select precision and optional table settings.
- For inverse solving, enter a target between −1 and 1.
- Press Calculate, then review steps and exact values.
- Copy, print, or export the completed results.
Worked Example
Suppose θ equals thirty degrees. Then 2θ equals sixty degrees, and sin(60°) equals √3/2. The identity produces the same exact final value correctly.
| Step | Expression | Result |
|---|---|---|
| 1 | θ | 30° |
| 2 | 2θ | 60° |
| 3 | sin(2θ) | sin(60°) = √3/2 ≈ 0.866025 |
| 4 | 2sin(θ)cos(θ) | 2 × 1/2 × √3/2 = √3/2 |
Common Special Angles
| θ | sin(θ) | cos(θ) | sin(2θ) | sin²(θ) |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | 1/2 | √3/2 | √3/2 | 1/4 |
| 45° | √2/2 | √2/2 | 1 | 1/2 |
| 60° | √3/2 | 1/2 | √3/2 | 3/4 |
| 90° | 1 | 0 | 0 | 1 |
Practical Uses
Double-angle relationships appear in waves, rotations, signals, and geometry. They simplify products and reveal repeated periodic patterns clearly. Engineers also use them when analysing oscillating system components.
Frequently Asked Questions
1. Is sin(2θ) the same as sin²(θ)?
No, these expressions calculate different quantities. Sin(2θ) doubles the angle before evaluating sine. Sin²(θ) squares the sine value after evaluation instead.
2. Which identity calculates sin(2θ)?
The standard identity is sin(2θ) = 2sin(θ)cos(θ). It works for degrees, radians, and gradians alike. Correct unit selection keeps every entered angle interpreted properly.
3. Can I enter several angles together?
Yes, enter values separated by commas or spaces. Semicolons and new lines are also accepted. The calculator processes up to fifty angles per submission.
4. Why are multiple inverse solutions shown?
Sine repeats regularly across its complete period. Therefore one target can match many different angles. The selected interval determines which valid solutions remain visible.
5. What values can sin(2θ) produce?
Its output always remains between negative one and one. Values outside that interval have no real-angle solution. The calculator validates this requirement before solving inverse equations.
6. Does the calculator show exact answers?
Yes, recognised special angles receive exact radical or fractional forms. Decimal approximations appear beside those exact values. Other angles receive accurate decimal values using selected precision.
7. How is the reference angle determined?
The entered angle is first normalised around one rotation. Its quadrant then determines the acute reference angle. This helps explain signs and special-angle relationships more clearly.
8. Can the identity be verified numerically?
Yes, choose the identity verification calculation mode. Both sides are evaluated independently for comparison. Their absolute difference reveals any floating-point rounding variation clearly.
9. Can I export the results?
Yes, results can be copied, printed, or downloaded. CSV exports support tables and spreadsheet analysis. PDF export creates a convenient document for later reference.