Half Angle Calculator
Find sin(θ/2), cos(θ/2) and tan(θ/2) for any angle.
Change the values and press Calculate to work out your own figures.
Enter an angle θ and choose degrees or radians.
Show the working
- sin(θ/2) = ±√((1 − cos θ) ÷ 2)
- cos(θ/2) = ±√((1 + cos θ) ÷ 2)
- tan(θ/2) = (1 − cos θ) ÷ sin θ; the signs follow the quadrant of θ/2
How to use it
Enter an angle θ and choose degrees or radians.
Enter the values you know (decimals, fractions, surds such as 2√5, or angles such as π/3 in radians). Show the working gives the rule used at each step.
Key facts
sin(θ/2) = ±√((1 − cos θ)/2); cos(θ/2) = ±√((1 + cos θ)/2)
Choosing the sign
The ± in the half-angle formulas is decided by the quadrant of θ/2; the calculator works with θ/2 directly, so the sign is always right.
Questions
What is sin(θ/2) for θ = 80°?
sin 40° ≈ 0.6427876097.
Exact value of sin 15°?
(√6 − √2)/4.
Formulas
sin(θ/2) = ±√((1 − cos θ)/2); cos(θ/2) = ±√((1 + cos θ)/2)
Sources
Limitations
- Angles can be typed in degrees or radians; radians may be written with π (π/3, 2pi/3).