Sin 2 Theta Calculator - Double Angle Formula
Enter an angle or a pair of trig values to compute sin(2θ) using the double angle identity sin(2θ) = 2 sin(θ) cos(θ). You also get cos(2θ) and tan(2θ), a step-by-step breakdown of the arithmetic, and an interactive chart showing how sin(2θ) behaves across a full cycle. Three input modes cover every scenario: a direct angle in degrees or radians, a known sin and cos pair, or a tangent value.
Formula
Worked example
For θ = 30°: sin(30°) = 0.5, cos(30°) = √3/2 ≈ 0.866025. Then sin(60°) = 2 × 0.5 × 0.866025 = 0.866025 = √3/2. cos(60°) = 0.866025² - 0.5² = 0.75 - 0.25 = 0.5. This matches the direct evaluation of sin(60°) and cos(60°).
What is the double angle formula for sine?
The double angle formula for sine states that sin(2θ) = 2 sin(θ) cos(θ). It follows from the compound angle identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B) when you set A = B = θ. The result is a single trig value for twice the angle, computed entirely from the sine and cosine of the original angle. Two alternative forms are sometimes more convenient: sin(2θ) = (sin(θ) + cos(θ))² - 1, and the tangent-based version sin(2θ) = 2tan(θ)/(1 + tan²(θ)). All three are algebraically equivalent; which one you reach for depends on what information is given.
How to calculate sin(2θ) step by step
If you know the angle θ directly, convert it to radians if needed, then evaluate sin(θ) and cos(θ) and multiply: sin(2θ) = 2 × sin(θ) × cos(θ). For θ = 45°, for example, sin(45°) = cos(45°) = √2/2 ≈ 0.7071, so sin(90°) = 2 × 0.7071 × 0.7071 = 1.0000. If you only know sin(θ) and not the angle itself, use the Pythagorean identity to find cos(θ) = ±√(1 - sin²(θ)), taking the sign from the quadrant. If you know tan(θ), rewrite sin(θ) = tan(θ)/√(1 + tan²(θ)) and cos(θ) = 1/√(1 + tan²(θ)) and substitute. The calculator handles all three paths automatically.
Period, range, and key properties of sin(2θ)
Because the argument is doubled, sin(2θ) completes one full cycle every 180° (π radians) instead of the usual 360° (2π). Its range remains [-1, 1] and its amplitude is 1. The function reaches its maximum of 1 at θ = 45° + n×180° and its minimum of -1 at θ = 135° + n×180°. It crosses zero at θ = 0°, 90°, 180°, 270°, and so on. In calculus, the derivative of sin(2θ) with respect to θ is 2cos(2θ), while the integral of sin(2θ) is -cos(2θ)/2 + C. These properties make the double angle formula central to simplifying integrals and Fourier series.
Where sin(2θ) appears in science and engineering
The double angle identity appears in projectile motion (the horizontal range R = v²sin(2θ)/g is maximised at 45°), in optics when computing intensity patterns from two interfering waves, and in electronics where sin(2ωt) terms arise naturally in amplitude and frequency modulation. In signal processing and Fourier analysis, every sinusoidal signal can be expressed as a sum of sin and cos terms; the double angle form frequently simplifies these decompositions. Structural engineers use it when resolving forces on inclined planes and in Mohr circle analysis for stress transformation.
Common sin(2θ) values at standard angles
| θ (degrees) | θ (radians) | sin(θ) | cos(θ) | sin(2θ) | Exact form |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | 0 |
| 15° | π/12 | 0.258819 | 0.965926 | 0.500000 | 1/2 |
| 30° | π/6 | 0.500000 | 0.866025 | 0.866025 | √3/2 |
| 45° | π/4 | 0.707107 | 0.707107 | 1.000000 | 1 |
| 60° | π/3 | 0.866025 | 0.500000 | 0.866025 | √3/2 |
| 75° | 5π/12 | 0.965926 | 0.258819 | 0.500000 | 1/2 |
| 90° | π/2 | 1 | 0 | 0 | 0 |
| 120° | 2π/3 | 0.866025 | -0.500000 | -0.866025 | -√3/2 |
| 135° | 3π/4 | 0.707107 | -0.707107 | -1.000000 | -1 |
| 150° | 5π/6 | 0.500000 | -0.866025 | -0.866025 | -√3/2 |
| 180° | π | 0 | -1 | 0 | 0 |
Exact or 6-decimal approximations for the most frequently used angles in trigonometry.
Frequently asked questions
What is sin(2θ) equal to?
sin(2θ) = 2 sin(θ) cos(θ). This is the double angle identity for sine. It is derived from the sum formula sin(A + B) by setting A = B = θ. The result always lies between -1 and 1, and it completes one full oscillation for every 180° change in θ.
How do I find sin(2θ) if I only know sin(θ)?
You also need cos(θ) or the quadrant. Use the Pythagorean identity: cos(θ) = ±√(1 - sin²(θ)). The sign depends on the quadrant: positive in Q1 and Q4, negative in Q2 and Q3. Once you have cos(θ), multiply: sin(2θ) = 2 sin(θ) cos(θ). Select the "sin and cos values" mode in this calculator and enter both values.
Is sin(2θ) the same as 2 sin(θ)?
No. sin(2θ) = 2 sin(θ) cos(θ), which equals 2 sin(θ) only when cos(θ) = 1, i.e., when θ = 0°. In general the two expressions differ. For example, sin(60°) ≈ 0.866, while 2 sin(30°) = 2 × 0.5 = 1.0, yet sin(2 × 30°) = sin(60°) ≈ 0.866.
What angle maximises sin(2θ)?
sin(2θ) reaches its maximum of 1 when 2θ = 90°, i.e., when θ = 45°. In projectile motion this corresponds to the launch angle that gives the greatest horizontal range on flat ground. Repeating maxima occur at 45° + 180°n for any integer n.
What is the period of sin(2θ)?
The period of sin(2θ) is 180° or π radians, exactly half the period of ordinary sin(θ). Multiplying the argument by 2 compresses the wave horizontally by a factor of two, so a full sine cycle fits inside 0° to 180° instead of 0° to 360°.
Can I use this calculator for radians?
Yes. In the "Angle θ" input mode, switch the unit dropdown from Degrees to Radians before entering your value. For example, enter θ = π/6 (approximately 0.5236) to compute sin(π/3) = √3/2 ≈ 0.866025. The sin/cos mode and tan mode are unit-independent since they accept dimensionless trig values directly.