Tangent Half Angle Calculator
This tangent half angle calculator finds t = tan(x/2), then rebuilds sin x and cos x through the Weierstrass substitution, a technique that turns trigonometric expressions into ordinary algebra. Enter an angle and choose degrees or radians, and the calculator returns the half angle tangent t, with sin x as 2t / (1 + t squared) and cos x as (1 - t squared) / (1 + t squared), so you can check both against the original angle. The substitution is a standard calculus tool: it converts integrals containing sin x and cos x into integrals of rational functions of t that can be solved with partial fractions, and it also appears in computer graphics and number theory. A worked example shows x = 60 degrees giving t = 0.577350, with the reconstructed sin x and cos x matching sin 60 and cos 60 exactly, confirming the tool before you apply it to your own angle. Bear in mind that t is undefined at odd multiples of 180 degrees, and that results are rounded for display. Use this calculator to check working by hand, explore how the substitution behaves, or save time converting an angle into its Weierstrass form.
The formula
With t = tan(x/2): sin x = 2t / (1 + t²), cos x = (1 − t²) / (1 + t²), and tan x = 2t / (1 − t²). These rational forms are the Weierstrass substitution.
Worked example
For x = 60 degrees, t = tan 30 = 0.577350. Then sin x = 2(0.577350) / (1 + 0.333333) = 0.866025 = sin 60, and cos x = (1 − 0.333333) / 1.333333 = 0.5 = cos 60. Enter 60 to confirm.
Frequently asked questions
What is the Weierstrass substitution?
Setting t = tan(x/2), so that sin x = 2t/(1+t squared) and cos x = (1 minus t squared)/(1 + t squared). It turns trig integrals into algebraic ones.
Why is it useful?
It converts an integral with sin x and cos x into an integral of a rational function of t, which can be solved with partial fractions.
When does t blow up?
When x is an odd multiple of 180 degrees, tan(x/2) is undefined; the substitution is used away from those points.
Who this calculator is for
This calculator is for calculus students and anyone using the tangent half angle substitution.
What this calculator assumes
- You enter one angle in the chosen unit.
- t = tan(x/2) is evaluated in radians.
- Results are rounded for display.
Formula and sources
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