This calculator finds sine, cosine and tangent at twice the angle you enter. An input of 30° therefore produces values for 60°. It does not simply multiply the three original trigonometric values by two.
Double-angle identities are useful when working from a known sine or cosine, simplifying expressions, or checking a trigonometry exercise. This form starts with an angle in degrees and evaluates the functions at the doubled angle. Results are rounded to five decimal places.
Keep the original angle separate from the target angle in your written work. Labeling both numbers prevents an accidental second doubling when you check the answer on another calculator.
| Quantity | Meaning |
|---|---|
| θ | The original angle, entered in degrees. There is no radians selector in this form. |
| 2θ | Twice the original angle. This is the angle used for the displayed sine, cosine and tangent. |
| Displayed ratios | Dimensionless values rounded to five decimal places. Sine and cosine are between −1 and 1; tangent has no such bound. |
Take θ = 30°. The target angle is 60°. Since sin(30°) = 1/2 and cos(30°) = √3/2, the sine identity gives 2 × (1/2) × (√3/2) = √3/2 ≈ 0.86603.
For cosine, (√3/2)² − (1/2)² = 3/4 − 1/4 = 1/2, displayed as 0.5. Dividing the unrounded sine by cosine gives tan(60°) = √3 ≈ 1.73205.
Now try θ = 45°. The target is 90°: sine is 1, cosine is 0 and tangent is undefined. A very large number from a different calculator near this angle can be a numerical approximation caused by a tiny cosine denominator.
For θ = 60°, the target angle is 120°. Its sine is √3/2, cosine is −1/2 and tangent is −√3. The sine stays positive but the other two values are negative because the target lies in the second quadrant. This example is useful when an answer has the right magnitude but the wrong sign.
If an exercise gives cos θ = 3/5, you can find cos(2θ) without solving for θ: 2(3/5)² − 1 = 18/25 − 25/25 = −7/25. Finding sin(2θ) would also require the sign of sin θ. The angle-based form needs an actual degree value, so keep this identity calculation in your written work rather than entering 3/5 as an angle.
These identities provide several ways to reach the same value. If you know only sin θ, use cos(2θ) = 1 − 2sin²θ; if you know only cos θ, use 2cos²θ − 1. This can save a separate calculation of the missing function.
To check a result, verify sin²(2θ) + cos²(2θ) ≈ 1. Small differences are expected if you use the rounded display. When cosine is not zero, sine divided by cosine should also agree with tangent within rounding error.
No. Sine is not a linear function of angle. The identity includes both sine and cosine of the original angle: sin(2θ) = 2sin θ cos θ.
The calculation uses twice the input, or 90°. Tangent is sine divided by cosine, and cosine is zero at 90°, so the ratio is undefined.
They are mathematically equivalent. Differences in a numerical check usually come from rounded intermediate values or using the wrong angle unit. Keep full precision until the final step.
The identities are exact mathematical relationships, but this implementation evaluates trigonometric functions numerically and rounds the output. An answer such as √3/2 and a displayed approximation of 0.86603 describe the same value at different levels of precision.
Adding 180° to θ adds 360° to the doubled angle, leaving its sine and cosine unchanged. Tangent at the doubled angle also repeats, apart from its undefined positions. Periodicity can help reduce a large degree input to a more familiar one, but reduce it using the target function’s period rather than dropping digits from the angle.
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