Mathematics · Advanced university
Sine coefficient from orthogonality · formula reference
This uses the real Fourier convention on [−π,π]. Distinct sine modes are orthogonal.
Formula
This uses the real Fourier convention on [−π,π]. Distinct sine modes are orthogonal.
bₙ = (1/π)∫₋π^π f(x)sin(nx) dx
Worked example
For f(x) = (-15)sin(2x) + 5sin(5x) on [−π,π], find the Fourier sine coefficient b_2, using b_j=(1/π) integral of f(x)sin(jx). Round the final result to 4 decimal places if needed.
Cross terms vanish by orthogonality. The integral of sin²(2x) is π, leaving coefficient -15.
-15
Prerequisites and checking
Algebra; calculus or linear algebra as appropriate to the topic. Keep units and rounding consistent. State the model conditions before applying the formula.
Further reading and review references
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