What are the half-angle formulae?
sin(A/2) = ±√((1−cosA)/2), cos(A/2) = ±√((1+cosA)/2), and tan(A/2) = (1−cosA)/sinA = sinA/(1+cosA). They come from rearranging the double-angle forms of cos 2θ.
// maths › Trigonometric Identities
Half-Angle Formulae
Evaluate sin(A/2), cos(A/2) and tan(A/2) from cos A using the half-angle formulae, with correct sign by quadrant and the t-formula link, in degrees or radians.
sin(A/2)=±√((1−cosA)/2); cos(A/2)=±√((1+cosA)/2); tan(A/2)=(1−cosA)/sinA
Angle unit
Frequently asked questions
How do I choose the ± sign?
By the quadrant the half-angle A/2 falls in. For example if A/2 is in the second quadrant, sin(A/2) is positive and cos(A/2) is negative. This calculator picks the sign from the actual half-angle.
Can you show a worked example?
For A = 60°, sin(A/2) = sin30° = √((1−cos60°)/2) = √((1−0.5)/2) = 0.5. In radians 60° = π/3 ≈ 1.0472 rad and A/2 = π/6; switch the toggle to see it that way.
What is the t-formula?
Writing t = tan(A/2) gives sinA = 2t/(1+t²), cosA = (1−t²)/(1+t²) and tanA = 2t/(1−t²). It turns trig equations into algebra and is key to integrating rational trig functions.
Where are they used?
The t-formula substitution in integration, solving certain trig equations, and deriving exact values for angles such as 15° and 22.5°.