Calculate sin(2θ) from an angle in degrees or radians, or from sin θ and cos θ, including quadrant-based values and step-by-step work.
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Sin 2 Theta Formula
The calculator uses one of two equivalent forms depending on what you enter.
sin(2θ) = 2 · sin(θ) · cos(θ)
When you enter the angle directly, it computes:
sin(2θ) = sin(2 · θ)
When you enter sin θ with a quadrant, it recovers cos θ from the Pythagorean identity:
cos(θ) = ± √(1 − sin²(θ))
- θ — the input angle, in degrees or radians
- sin θ — sine of θ, between −1 and 1
- cos θ — cosine of θ, between −1 and 1
- sin(2θ) — the result, always between −1 and 1
The sign of cos θ is set by the quadrant: positive in Q I and Q IV, negative in Q II and Q III. If you enter sin θ and cos θ that do not satisfy sin²θ + cos²θ = 1, the calculator flags it but still applies the double-angle identity to your inputs.
Common Values and Sign Reference
Exact values of sin(2θ) at standard angles:
| θ | 2θ | sin(2θ) |
|---|---|---|
| 0° | 0° | 0 |
| 15° | 30° | 1/2 |
| 30° | 60° | √3/2 ≈ 0.8660 |
| 45° | 90° | 1 |
| 60° | 120° | √3/2 ≈ 0.8660 |
| 75° | 150° | 1/2 |
| 90° | 180° | 0 |
| 135° | 270° | −1 |
Sign of sin(2θ) by the quadrant of θ:
| Quadrant of θ | sin θ | cos θ | sin(2θ) |
|---|---|---|---|
| I | + | + | + |
| II | + | − | − |
| III | − | − | + |
| IV | − | + | − |
Worked Example
Given sin θ = 3/5 and θ in Quadrant II, find sin(2θ).
- cos θ = ±√(1 − 9/25) = ±4/5. In Q II, cos θ is negative, so cos θ = −4/5.
- sin(2θ) = 2 · (3/5) · (−4/5) = −24/25 = −0.96.
FAQ
Is sin(2θ) the same as 2·sin(θ)? No. sin(2θ) = 2·sin(θ)·cos(θ). The two only match when cos θ = 1, that is, at θ = 0, ±2π, and so on.
What is the maximum value? sin(2θ) reaches 1 when 2θ = 90° + 360°k, so θ = 45°, 225°, and so on.
Does the calculator accept negative angles? Yes. Enter any real number in degrees or radians. sin(2θ) is an odd function, so sin(−2θ) = −sin(2θ).
Why does it warn when sin²θ + cos²θ ≠ 1? Those two inputs must come from the same angle. If they do not satisfy the Pythagorean identity, your inputs are inconsistent and the result will not match any real angle.
