Triple Angle Identities

Function	Formula	Description
sin(3θ)	$3\sin\theta - 4\sin^3\theta$	Cubic polynomial in sine - derived from angle addition formulas
cos(3θ)	$4\cos^3\theta - 3\cos\theta$	Cubic polynomial in cosine - fundamental for Chebyshev polynomials
tan(3θ)	$\displaystyle\frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$	Rational function - undefined when 3tan²θ = 1, i.e., tanθ = ±√3/3
csc(3θ)	$\displaystyle\frac{\csc\theta}{3 - 4\sin^2\theta}$	Reciprocal form involving original cosecant - undefined when sin(3θ) = 0
sec(3θ)	$\displaystyle\frac{\sec\theta}{4\cos^2\theta - 3}$	Reciprocal form involving original secant - undefined when cos(3θ) = 0
cot(3θ)	$\displaystyle\frac{\cot^3\theta - 3\cot\theta}{3\cot^2\theta - 1}$	Rational function in cotangent - undefined when tan(3θ) = 0