What is the formula of sin2x

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The sin2x formulas are fundamental trigonometric identities that appear frequently in mathematics, physics, and engineering. This comprehensive guide covers all essential formulas related to sin2x, including double angle formulas, power reduction formulas, and various transformations.

Complete Sin2x Formula Table

Formula Type	Formula	Alternative Forms	Description	Common Use
Double Angle Formula	sin(2x) = 2sin(x)cos(x)	sin(2θ) = 2sin(θ)cos(θ)	Primary double angle formula for sine	Finding sin of double angles
Sin2x in Terms of Tangent	sin(2x) = 2tan(x)/(1 + tan²(x))	sin(2θ) = 2tan(θ)/(1 + tan²(θ))	Expresses sin2x using only tangent	When tan(x) is known
Power Reduction (Sin²x)	sin²(x) = (1 - cos(2x))/2	sin²(θ) = (1 - cos(2θ))/2	Reduces power of sine	Integration and simplification
Power Expansion (Sin²(2x))	sin²(2x) = (1 - cos(4x))/2	sin²(2θ) = (1 - cos(4θ))/2	Power reduction for sin²(2x)	Advanced integration
Product Formula	sin(2x)cos(2x) = sin(4x)/2	sin(2θ)cos(2θ) = sin(4θ)/2	Product of sin2x and cos2x	Simplifying products
Sum/Difference with Cos2x	sin(2x) + cos(2x) = √2 sin(2x + π/4)	sin(2θ) + cos(2θ) = √2 sin(2θ + π/4)	Combines sin2x and cos2x	Harmonic analysis
Sum/Difference with Cos2x	sin(2x) - cos(2x) = √2 sin(2x - π/4)	sin(2θ) - cos(2θ) = √2 sin(2θ - π/4)	Difference of sin2x and cos2x	Wave analysis
Triple Angle Relation	sin(3x) = 3sin(x) - 4sin³(x) = sin(x)(3 - 4sin²(x))	sin(3θ) = 3sin(θ) - 4sin³(θ)	Related triple angle formula	Advanced trigonometry
Inverse Formula	x = (1/2)arcsin(y/2) where y = sin(2x)	θ = (1/2)arcsin(y/2)	Finding x when sin(2x) is known	Solving equations
Half Angle in Terms of 2x	sin(x) = ±√((1 - cos(2x))/2)	sin(θ) = ±√((1 - cos(2θ))/2)	Half angle using double angle	Deriving other formulas

Cos2x Formulas (Related to Sin2x)

Formula	Description
cos(2x) = cos²(x) - sin²(x)	Basic double angle for cosine
cos(2x) = 2cos²(x) - 1	Alternative form using cos²
cos(2x) = 1 - 2sin²(x)	Alternative form using sin²
cos(2x) = (1 - tan²(x))/(1 + tan²(x))	In terms of tangent

Combined Sin2x and Cos2x Identities

Identity	Formula	Application
Pythagorean	sin²(2x) + cos²(2x) = 1	Fundamental identity
Product-to-Sum	sin(2x)cos(2x) = (1/2)sin(4x)	Converting products
Tangent Double	tan(2x) = sin(2x)/cos(2x) = 2tan(x)/(1-tan²(x))	Double angle tangent

What is the formula of sin2x

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