Half Angle Calculator

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What is Half Angle?

A half-angle calculation finds sine, cosine and tangent for half of a known angle. Entering 60°, for example, asks for the trigonometric values at 30°. The answer is a set of ratios, not simply the number 30.

This page uses square-root identities and reports nonnegative square roots. That detail matters outside the usual classroom examples: it gives magnitudes, and does not choose the correct sign for every possible quadrant. Use the result directly for an original angle strictly between 0° and 180°; otherwise check the quadrant and any undefined values separately.

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How to Use This Calculator

  1. Enter the original angle θ in degrees, before halving it.
  2. Choose Calculate. The three results correspond to sin(θ/2), cos(θ/2) and tan(θ/2).
  3. Work out θ/2 yourself and check which quadrant contains it.
  4. Keep any exact radical form in written work if the question asks for an exact answer. The displayed values are rounded to four decimal places.

The labels Sin/2, Cos/2 and Tan/2 on the result panel refer to functions of the half-angle. They do not mean that an ordinary sine, cosine or tangent value has been divided by two.

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Formula Used

sin(θ/2) = ±√((1 − cos θ)/2); cos(θ/2) = ±√((1 + cos θ)/2); tan(θ/2) = ±√((1 − cos θ)/(1 + cos θ)). This implementation takes the nonnegative roots; the full identities require quadrant-dependent signs.
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Variables Explained

QuantityMeaning
θThe original angle in degrees, not the half-angle. The calculation converts degrees internally for the cosine operation.
θ/2The angle whose trigonometric values you want.
Sine, cosine and tangent resultsDimensionless ratios. Their signs depend on the quadrant of θ/2, even though this implementation returns nonnegative roots.
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Worked Example

Enter θ = 120°. The angle you are investigating is 60°, and cos(120°) = −0.5. The sine calculation becomes √((1 − (−0.5))/2) = √0.75 ≈ 0.8660. Cosine becomes √((1 + (−0.5))/2) = √0.25 = 0.5000.

Tangent is √((1 − (−0.5))/(1 + (−0.5))) = √3 ≈ 1.7321. All three signs are positive because 60° lies in the first quadrant. As a check, sine divided by cosine is approximately 1.7321.

Finding a less familiar angle

To investigate 15°, start with θ = 30°. Using cos(30°) = √3/2, the sine identity becomes √((1 − √3/2)/2), which is equivalent to (√6 − √2)/4. Its decimal value is about 0.2588. Cosine similarly becomes (√6 + √2)/4, or about 0.9659. These exact forms show why an angle that is not in the usual short table can still have an exact trigonometric expression.

When checking this example, do not replace √3/2 with 0.87 before substituting. That rounding changes the quantity inside a square root and can noticeably affect the final digits. Substitute exact values first, simplify where possible, then compare the rounded decimal with the page.

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Why Use This Calculator?

Half-angle identities are useful when you know the cosine of an angle but need trigonometric values for an angle half as large. They also help turn angles such as 15° and 22.5° into exact expressions involving familiar angles.

For comparison, if you enter 300°, the half-angle is 150°. Its cosine and tangent should be negative. The nonnegative results on this page need that sign correction; do not treat them as signed values for general angles. At a half-angle of 90°, tangent is undefined, so a finite tangent display should not be used.

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Common Mistakes

    • Do not enter the half-angle when the form asks for the original angle. To investigate 30°, enter 60°.
    • Use degrees rather than radians in this form.
    • Do not assume a square root supplies a quadrant sign. The principal square root is nonnegative.
    • If 1 + cos θ is zero, tangent has a zero denominator. A displayed zero or rounded finite number is not a valid substitute for an undefined result.
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Frequently Asked Questions

Why is sin(θ/2) different from sin(θ)/2?

The first halves the angle before applying sine. The second halves the sine value afterwards. With θ = 60°, these are 0.5 and about 0.4330, respectively.

Why might another calculator show a negative answer?

It may evaluate the signed trigonometric function directly. This page uses nonnegative square-root expressions. Check the quadrant of the half-angle before comparing answers.

Can I use the displayed decimals as exact answers?

No. Four-place decimals are rounded approximations. For an exact exercise, retain a radical such as √3/2 and round only when the question requests a decimal.

Does halving an angle change its unit?

No. Half of 120° is 60°, still in degrees. If the original angle were expressed in radians, the half-angle would remain in radians, but this particular form expects degrees. Convert radians to degrees before using it.

Can I check the sine and cosine outputs together?

For a valid real half-angle, sin²(θ/2) + cos²(θ/2) = 1. Squaring the displayed results and adding them should give approximately one. This checks magnitudes but cannot confirm quadrant signs, because squaring removes a minus sign. Use both the identity and a separate quadrant check when evaluating angles outside the first quadrant.

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Sources & References

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