Key takeaways
- sin 2θ = 2 sin θ cos θ.
- It uses the ORIGINAL angle's sine and cosine, not the doubled angle's.
- sin 2θ is NOT the same as 2 sin θ — you must also multiply by cos θ.
- It comes from the sine addition formula with both angles set equal to θ.
In this guide
What is the double angle formula for sine?
It's a shortcut identity from trigonometry that expresses sin(2θ) in terms of sin θ and cos θ, instead of needing to find sin(2θ) from scratch. It's especially useful in calculus and physics, where angles often need to be doubled mid-calculation.
Why does the double angle formula matter?
Recalculating sin(2θ) from scratch every time wastes work you've already done. If you already know sin θ and cos θ, this identity gets you sin(2θ) with one multiplication instead of a fresh trig lookup.
sin 2θ is not simply double sin θ — that's a common trap. The correct relationship needs cos θ multiplied in too, which is exactly what makes the formula worth memorizing rather than guessing.
The formula
| Symbol | What it means | Example |
|---|---|---|
| θ (theta) | The original angle, in degrees | 20° |
| sin θ, cos θ | Sine and cosine of the original angle | 0.3420, 0.9397 |
| sin 2θ | Sine of the doubled angle | 0.6428 |
How to find sin 2θ step by step
- 1Find sin θ and cos θ for your original angle. Use the sin, cos, tan calculator if needed.
- 2Multiply sin θ by cos θ.
- 3Multiply that result by 2.
- 4The answer is sin(2θ), the sine of the doubled angle.
Enter your own angle below to see sin 2θ calculated step by step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using the doubled angle's own sine and cosine. The formula only needs the original angle's sine and cosine, not the doubled angle's.
- Forgetting the angle doubles too. If θ = 30°, the result is
sin(60°), not sin(30°) doubled. - Mixing up degrees and radians when finding sin θ and cos θ in the first place.
Tips and tricks
- It pairs naturally with the double angle formula for cosine, which uses the same original angle to find cos(2θ) instead.
- Sanity-check with θ = 45°:
sin(90°) = 1, and2 × sin 45° × cos 45° = 2 × 0.7071 × 0.7071 = 1matches exactly. - The result should always stay between -1 and 1, since it's still a sine value.
Where you'll use it in real life
- Calculus: simplifying integrals and derivatives that involve
sin(2θ). - Physics: analyzing wave interference and oscillation problems.
- Engineering: signal processing calculations that double a frequency's phase angle.
- Electronics: power calculations in AC circuits, which often involve doubled-angle sine terms.
Quick summary
sin 2θ = 2 sin θ cos θ reuses the sine and cosine of the original angle to find the sine of double that angle. It's not simply doubling sin θ — the cos θ factor is essential. Try the calculator above with your own angle.
Worked example
An engineer already calculated sin(20°) and cos(20°) for a signal analysis and now needs sin(40°) without recalculating from scratch.
- 01sin 2θ = 2 sin θ cos θ
- 02= 2 × sin 20° × cos 20°
- 03= 2 × 0.342 × 0.9397
- 04sin 40° = 0.6428
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the double angle formula for sine?
sin 2θ = 2 sin θ cos θ. It gives you the sine of double an angle using only the sine and cosine of the original angle.
Is sin 2θ the same as 2 sin θ?
No, that's a common mistake. You must multiply by cos θ as well: sin 2θ = 2 sin θ cos θ, not just 2 sin θ.
What is sin(2θ) when θ = 45°?
sin(90°) = 1, which matches 2 × sin(45°) × cos(45°) = 2 × 0.7071 × 0.7071 = 1.
Where does the double angle formula come from?
It comes from the sine addition formula, sin(A + B) = sin A cos B + cos A sin B, with both A and B set equal to θ.
