Double Angle Identity for Sin
sin2x=2sinxcosx
The double angle identity for sin is a trigonometric identity that relates the sine of twice an angle to the sine of the angle itself. The identity is given by:
sin(2θ) = 2sin(θ)cos(θ)
where θ represents any angle.
This identity can be derived by using the angle sum formula for sine:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
If we let a = b = θ, we get:
sin(2θ) = sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ)
Simplifying the right-hand side of the equation gives:
sin(2θ) = 2sin(θ)cos(θ)
This identity is useful in evaluating trigonometric expressions involving sine, as it allows us to express sin(2θ) in terms of sin(θ) and cos(θ) which can be easier to work with. For example, we can use this identity to find the value of sin(60°) by using the fact that 60° = 2(30°), and applying the identity we get:
sin(60°) = sin(2(30°)) = 2sin(30°)cos(30°) = 2(1/2)(√3/2) = √3/2
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