Math
Half Angle Formula Calculator
The half-angle formulas give sin(θ/2), cos(θ/2), and tan(θ/2) from cos θ alone, but each square root needs a sign that depends on where θ/2 lands.
Enter the angle, or cos θ (or sin θ) with the quadrant of θ, and the calculator picks the sign, keeps radicals exact, and writes out every step.
Fractions, decimals, and radicals work. Must be between -1 and 1.
θ/2 lands in Quadrant I when θ is in I or II, and in Quadrant II when θ is in III or IV.
Try an example
Result
sin(θ/2)
—
- cos(θ/2)
- —
- tan(θ/2)
- —
- θ/2
- —
- Quadrant of θ/2
- —
- Signs used
- —
- cos θ
- —
More details (1 more)
- sin θ
- —
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Study path
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The half-angle identities
They are the double-angle cosine formulas solved for the half angle. The tangent forms without ± are usually easier because they never need a sign decision.
Choosing the sign
| θ is in | θ/2 is in | sin(θ/2) | cos(θ/2) |
|---|---|---|---|
| Quadrant I (0° to 90°) | 0° to 45°, Quadrant I | + | + |
| Quadrant II (90° to 180°) | 45° to 90°, Quadrant I | + | + |
| Quadrant III (180° to 270°) | 90° to 135°, Quadrant II | + | − |
| Quadrant IV (270° to 360°) | 135° to 180°, Quadrant II | + | − |
How to use it
- Choose whether you know θ itself or a ratio of θ.
- For a ratio, enter its value and the quadrant of θ.
- Read sin(θ/2), cos(θ/2), tan(θ/2) with the signs already applied, then open the steps to see why.
How to read the answer
- When cos θ is a fraction, the half-angle values come out as exact A root expression written with the √ symbol, such as a square root. such as 2√5/5.
- When θ/2 is a special angle (θ = 30° gives 15°), The answer kept as a fraction or radical instead of a rounded decimal. like (√6 − √2)/4 are shown.
- tan(θ/2) is undefined only when cos θ = -1 (θ = 180°), where θ/2 = 90°.
Common mistakes and edge cases
- Using the quadrant of θ for the sign instead of the quadrant of θ/2.
- Dividing the cosine by 2: cos(θ/2) is not (cos θ)/2.
- Forgetting to rationalize: √(1/5) should be written √5/5.
- Entering cos θ positive for a Quadrant II angle: cosine is negative there, and the calculator flags the conflict.
Worked examples
cos θ = -3/5 in Quadrant III
sin(θ/2)
2√5/5 (0.894427)
cos θ = 7/25 in Quadrant IV
sin(θ/2)
3/5 (0.6)
sin θ = 4/5 in Quadrant II
sin(θ/2)
2√5/5 (0.894427)
Exact value of sin 15°
θ = 30°, so θ/2 = 15°
sin(θ/2)
(√6 − √2)/4 (0.258819)
θ = 120°
θ/2 = 60°
sin(θ/2)
√3/2 (0.866025)
Angle past 360°
θ = 400°, θ/2 = 200° in Quadrant III
sin(θ/2)
-0.34202
θ = 180°
tan(θ/2) undefined
sin(θ/2)
1
Sign conflict
sin(θ/2)
—
Frequently asked questions
How do I know which sign to use in the half-angle formula?+
Find the quadrant of θ/2, not θ. If θ is between 180° and 360°, θ/2 is between 90° and 180° (Quadrant II), so sin(θ/2) is positive and cos(θ/2) is negative.
What is the half-angle formula for tangent?+
tan(θ/2) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ. These forms need no ± sign, which makes them the safest choice.
How do I find the exact value of sin 15°?+
Use θ = 30°: sin 15° = √((1 − cos 30°)/2) = √((1 − √3/2)/2) = √(2 − √3)/2, which equals (√6 − √2)/4 ≈ 0.2588.
Why does the calculator ask for the quadrant of θ?+
Because cos θ alone does not tell you where θ/2 is. cos θ = 1/2 could be θ = 60° (θ/2 = 30°) or θ = 300° (θ/2 = 150°), which have different signs for cos(θ/2).
Can I use the half-angle formula with sin θ instead of cos θ?+
Yes. First find cos θ from cos²θ = 1 − sin²θ with the sign from the quadrant, then apply the formulas. The sin θ mode does exactly that.
Is cos(θ/2) equal to (cos θ)/2?+
No. For θ = 60°, cos 30° = √3/2 ≈ 0.866 while (cos 60°)/2 = 0.25. Only the half-angle formula gives the right value.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026