Doubling an angle combines one rotation with an identical second rotation.
WHAT HAPPENS IF…At θ=90°, what is sin(2θ)?
Interactive exploration
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Connect the picture to the ideasin2θ=2sinθcosθ,cos2θ=cos2θ−sin2θ
Set α=β=θ in the angle-addition formulas. The sine terms become identical, giving twice their product. Using sin²θ+cos²θ=1 also gives cos2θ=1−2sin²θ=2cos²θ−1.
A detail that matters
A doubled angle does not generally double a trigonometric function value.