Triple Angle Identities: From Intuition to Formula
You already know how sin2θ relates to sinθ — a double-angle identity. The triple-angle identities go one step further: they express sin3θ, cos3θ, and tan3θ using only sinθ, cosθ, or tanθ.
The core idea: a triple angle is just a double angle plus the original, 3θ=2θ+θ. So everything follows from the sum formulas you already know.
The Precise Statements
Triple Angle Identities
sin3θ=3sinθ−4sin3θ
cos3θ=4cos3θ−3cosθ
tan3θ=1−3tan2θ3tanθ−tan3θ
Where do they come from?
Deriving sin3θ
Start with sin(2θ+θ):
sin3θ=sin2θcosθ+cos2θsinθ
Replace sin2θ=2sinθcosθ and cos2θ=1−2sin2θ (this form keeps everything in sinθ):
sin3θ=(2sinθcosθ)cosθ+(1−2sin2θ)sinθ=2sinθcos2θ+sinθ−2sin3θ
Now use cos2θ=1−sin2θ:
sin3θ=2sinθ(1−sin2θ)+sinθ−2sin3θ=2sinθ−2sin3θ+sinθ−2sin3θ=3sinθ−4sin3θ
The −4sin3θ comes from combining −2sin3θ and −2sin3θ — the most common place for an arithmetic slip.
Deriving cos3θ
Start with cos(2θ+θ):
cos3θ=cos2θcosθ−sin2θsinθ
Use cos2θ=2cos2θ−1 and sin2θ=2sinθcosθ:
cos3θ=(2cos2θ−1)cosθ−2sin2θcosθ=2cos3θ−cosθ−2sin2θcosθ
Replace sin2θ=1−cos2θ:
cos3θ=2cos3θ−cosθ−2(1−cos2θ)cosθ=4cos3θ−3cosθ
Deriving tan3θ
Use tan(A+B) with A=2θ, B=θ:
tan3θ=1−tan2θtanθtan2θ+tanθ
With tan2θ=1−t22t where t=tanθ, multiply numerator and denominator by 1−t2:
tan3θ=(1−t2)−2t22t+t(1−t2)=1−3t23t−t3
What to Remember for Exams …