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Trigonometric Power Formulas

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Trigonometric power formulas express powers of sine and cosine in terms of multiple-angle trigonometric functions. Examples include

sin^2x=1/2[1-cos(2x)]
(1)
sin^3x=1/4[3sinx-sin(3x)]
(2)
sin^4x=1/8[3-4cos(2x)+cos(4x)]
(3)

and

cos^2x=1/2[1+cos(2x)]
(4)
cos^3x=1/4[3cosx+cos(3x)]
(5)
cos^4x=1/8[3+4cos(2x)+cos(4x)]
(6)

(Beyer 1987, p. 140). Formulas of these types can also be given analytically as

sin^(2n)x=1/(2^(2n))(2n; n)+((-1)^n)/(2^(2n-1))sum_(k=0)^(n-1)(-1)^k(2n; k)cos[2(n-k)x]
(7)
sin^(2n+1)x=((-1)^n)/(4^n)sum_(k=0)^(n)(-1)^k(2n+1; k)sin[(2n+1-2k)x]
(8)
cos^(2n)x=1/(2^(2n))(2n; n)+1/(2^(2n-1))sum_(k=0)^(n-1)(2n; k)cos[2(n-k)x]
(9)
cos^(2n+1)x=1/(4^n)sum_(k=0)^(n)(2n+1; k)cos[(2n+1-2k)x],
(10)

where n is a nonnegative integer and (n; m) is a binomial coefficient.

Additional useful power identities include

a^2cos^2x+b^2sin^2x=1/2[a^2+b^2+(a^2-b^2)cos(2x)] a^2cos^4x-b^2sin^4x =1/8{4(a^2+b^2)cos(2x)+(a^2-b^2)[cos(4x)+3]}.
(11)

See also

Trigonometric Addition Formulas, Trigonometry

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.

Referenced on Wolfram|Alpha

Trigonometric Power Formulas

Cite this as:

Weisstein, Eric W. "Trigonometric Power Formulas." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrigonometricPowerFormulas.html

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