What is the Fourier sine series?

The Fourier sine series represents an odd function using only sine terms: f(t) = Σ bₙsin(nω₀t). All cosine coefficients aₙ = 0 and the average a₀ = 0 due to antisymmetry. Classic examples include the square wave and sawtooth wave. Half-range sine expansions are used for Dirichlet (zero-value) boundary conditions.

When does a Fourier series have only cosine or only sine terms?

Only cosine terms appear when f(t) is even: f(−t) = f(t). Only sine terms appear when f(t) is odd: f(−t) = −f(t). If the function has neither symmetry, both cosine and sine terms are present. Half-wave symmetry f(t+T/2) = −f(t) eliminates even harmonics regardless of even/odd symmetry.

How do you convert between cosine-sine and amplitude-phase form?

Use cₙ = √(aₙ²+bₙ²) for the amplitude and φₙ = arctan(−bₙ/aₙ) for the phase. Then aₙcos(nω₀t) + bₙsin(nω₀t) = cₙcos(nω₀t + φₙ). The amplitude-phase form is preferred for spectral plots since |cₙ| shows harmonic strength regardless of phase, providing clearer frequency-domain visualization.

Fourier Series Cosine and Sine

Quick Answer

The Fourier series decomposes a periodic function into cosine and sine components: f(t) = a₀/2 + Σ[aₙcos(nω₀t) + bₙsin(nω₀t)], where cosine terms capture even symmetry and sine terms capture odd symmetry. Even functions f(t) = f(−t) have only cosine terms (Fourier cosine series, bₙ = 0), while odd functions f(−t) = −f(t) have only sine terms (Fourier sine series, aₙ = 0). The combined amplitude of each harmonic is cₙ = √(aₙ² + bₙ²) with phase φₙ = arctan(−bₙ/aₙ).

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Fourier Series Cosine and Sine Components

The trigonometric Fourier series represents a periodic function as a sum of cosines and sines at integer multiples of the fundamental frequency: f(t) = a₀/2 + Σ_{n=1}^∞ [aₙcos(nω₀t) + bₙsin(nω₀t)]. The cosine terms cos(nω₀t) are even functions (symmetric about t = 0), while the sine terms sin(nω₀t) are odd functions (antisymmetric). The coefficient aₙ measures how much of f(t) projects onto cos(nω₀t), and bₙ measures the projection onto sin(nω₀t). Together, each harmonic pair aₙcos(nω₀t) + bₙsin(nω₀t) can be written as cₙcos(nω₀t + φₙ), where cₙ = √(aₙ² + bₙ²) is the amplitude and φₙ = arctan(−bₙ/aₙ) is the phase. This decomposition connects directly to the Laplace transform framework at www.lapcalc.com.

Key Formulas

Fourier Cosine Series: Even Functions

When f(t) is an even function (f(−t) = f(t)), all sine coefficients vanish: bₙ = 0 for all n. The series reduces to the Fourier cosine series: f(t) = a₀/2 + Σ aₙcos(nω₀t). This occurs because the product of an even function f(t) with an odd function sin(nω₀t) is odd, and the integral of an odd function over a symmetric interval is zero. Examples include f(t) = |t| (triangle wave), f(t) = cos²(ω₀t), and any function defined only for t > 0 that is extended as an even function to negative t. The half-range cosine expansion applies the Fourier cosine series to functions defined on [0, L] by assuming even symmetry, useful for solving boundary value problems with Neumann conditions (zero-derivative boundaries).

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Fourier Sine Series: Odd Functions

When f(t) is odd (f(−t) = −f(t)), all cosine coefficients vanish: aₙ = 0 for all n, including a₀ = 0 (the average of an odd function is zero). The series becomes the Fourier sine series: f(t) = Σ bₙsin(nω₀t). The square wave (alternating ±A) is the classic example: f(t) = (4A/π)Σ sin(nω₀t)/n for odd n. The sawtooth wave also has a pure sine series. The half-range sine expansion represents a function on [0, L] by assuming odd symmetry, appropriate for boundary value problems with Dirichlet conditions (zero-value boundaries) such as vibrating strings fixed at both ends and heat conduction with fixed-temperature endpoints.

Amplitude-Phase Form and Harmonic Analysis

Converting between the cosine-sine form and amplitude-phase form: aₙcos(nω₀t) + bₙsin(nω₀t) = cₙcos(nω₀t + φₙ), where cₙ = √(aₙ² + bₙ²) and φₙ = −arctan(bₙ/aₙ). The magnitude spectrum |cₙ| versus n shows the strength of each harmonic regardless of phase, providing a concise frequency-domain characterization. In electrical engineering, total harmonic distortion (THD) measures the ratio of harmonic power to fundamental power: THD = √(Σ_{n≥2} cₙ²)/c₁. Power system standards (IEEE 519) limit THD to 5% for voltage and 5–20% for current. Vibration analysis uses the harmonic series to distinguish fault types: unbalance produces 1× fundamental, misalignment produces strong 2× harmonic, and looseness produces many high-order harmonics.

Complex Exponential Form: Unifying Cosine and Sine

Euler's formula e^(jθ) = cos(θ) + j·sin(θ) unifies the separate cosine and sine terms into a single complex exponential: f(t) = Σ_{n=−∞}^∞ cₙ·e^(jnω₀t), where cₙ = (aₙ − jbₙ)/2 for n > 0 and c₋ₙ = cₙ* (complex conjugate) for real signals. This exponential form is mathematically more compact and connects directly to the Fourier transform (let T → ∞) and the Laplace transform (replace jω with s). The complex coefficients encode both amplitude and phase in a single number: |cₙ| = cₙ/2 (half the amplitude, since positive and negative frequencies each contribute half) and ∠cₙ = φₙ. Engineering spectral analysis tools typically display the one-sided spectrum 2|cₙ| to show physical amplitudes. Compute the underlying transform operations at www.lapcalc.com.

Frequently Asked Questions

The Fourier cosine series represents an even function using only cosine terms: f(t) = a₀/2 + Σ aₙcos(nω₀t). All sine coefficients bₙ = 0 due to symmetry. It applies to any even function f(−t) = f(t) and to half-range expansions on [0,L] assuming even extension, used in boundary value problems with Neumann (zero-derivative) conditions.

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