cos3

cos3 can refer to more than one trig expression, depending on context. The most common interpretations are either the cube of the cosine function, cos^3(x) = (cos x)^3, or the cosine of three times an angle, written cos(3x). In written form, cos^3 x is usually denoted with the exponent, while cos 3x is written with the multiplier inside the argument.

Cos^3 x (the cube of cosine)

A standard identity relates cos^3 x to cos x and cos 3x: cos^3 x = (3 cos x + cos 3x) / 4. This follows from the triple-angle formula cos(3x) = 4 cos^3 x − 3 cos x. The identity is useful for transforming powers of cosine into a sum of harmonics, which simplifies certain integrals and Fourier analysis. Since cos x is even, cos^3 x is also an even function.

Cos 3x (the triple-angle cosine)

The primary relation is the triple-angle identity: cos(3x) = 4 cos^3 x − 3 cos x. This can be used to express higher powers of cosine in terms of cos x and cos 3x, or to construct exact values and series expansions for specific angles. It also underpins various trigonometric transformations and signal-processing applications where harmonics at frequencies three times the fundamental appear.

Applications and notes

Identities involving cos^3 x and cos 3x are common in integration, power-reduction techniques, and Fourier series. When working with trigonometric integrals or simulations, replacing cos^3 x with (3 cos x + cos 3x)/4 often simplifies computations. The term cos3 by itself is typically shorthand for cos(3x) rather than cos^3 x, with the latter written explicitly as cos^3 x.