How this calculator works
An identity holds for every angle in its domain, while an equation may hold only for particular angles. This tool checks representative sine identities and displays the six functions of θ for context. A small non-zero residual comes from floating-point arithmetic. Reflection and shift selectors make the transformation explicit. The squared half-angle form avoids an unresolved sign choice.
Formula
sin²θ+cos²θ=1; sin(-θ)=-sin θ; sin(θ+π/2)=cos θ; sin(3θ)=3sin θ-4sin³θ; sin²(θ/2)=(1-cos θ)/2.
Worked example
At θ = 30°, sin²θ+cos²θ = 1/4+3/4 = 1. The triple-angle identity gives sin 90° = 3(1/2)-4(1/2)³ = 1.
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Frequently asked questions
Does a zero residual prove an identity?
No. It confirms the equality only for the chosen inputs within numerical precision. A proof must establish the equality across its domain.
Why might the residual be tiny instead of zero?
Computer sine, cosine and arithmetic have limited precision. Values near 10⁻¹⁵ are ordinary rounding differences for moderate inputs.
Does the half-angle family select a quadrant?
It checks the squared identity, so both possible signs give the same value. Use the half-angle calculator for signed results.