Trigonify

Topics · 3 identities

Half Angle

Half-angle formulas for sine, cosine, and tangent of θ/2.

Half angle formula: sin(θ/2) = ±√((1 − cos θ)/2)

sin⁡θ2=±1−cos⁡θ2\sin\tfrac{\theta}{2} = \pm\sqrt{\dfrac{1 - \cos\theta}{2}}

sin(θ/2) = ±√((1 − cos θ)/2), obtained by rearranging cos 2θ = 1 − 2sin²θ. The ± is not optional: you choose the sign from the quadrant θ/2 lands in, not from the sign of θ. Forgetting to check that quadrant is the classic way to lose the mark.

Open identity

Half angle formula: cos(θ/2) = ±√((1 + cos θ)/2)

cos⁡θ2=±1+cos⁡θ2\cos\tfrac{\theta}{2} = \pm\sqrt{\dfrac{1 + \cos\theta}{2}}

cos(θ/2) = ±√((1 + cos θ)/2), from rearranging cos 2θ = 2cos²θ − 1. Note the plus inside the radical, against the minus in the sine version. As with sine, the sign in front is decided by the quadrant of θ/2.

Open identity

Half angle formula: tan(θ/2), both radical-free forms

tan⁡θ2=1−cos⁡θsin⁡θ=sin⁡θ1+cos⁡θ\tan\tfrac{\theta}{2} = \dfrac{1 - \cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1 + \cos\theta}

tan(θ/2) = (1 − cos θ)/sin θ = sin θ/(1 + cos θ). Unlike the sine and cosine half-angle formulas these need no ± and no square root - the sign takes care of itself. Use the first form when sin θ is comfortably non-zero and the second when cos θ is near −1.

Open identity