Halve an angle in degrees and calculate sine, cosine, and tangent with the correct quadrant signs.
Complete the required fields and select the calculate button.
Enter degrees, not radians. Tangent is undefined whenever the half-angle has cosine zero, such as 90° plus a multiple of 180°.
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 page also displays the half-angle itself.
The calculator evaluates the signed trigonometric functions at θ/2, so negative results are retained in quadrants where they belong. Tangent is reported as undefined when the half-angle has cosine zero.
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.
| Quantity | Meaning |
|---|---|
| θ | The original angle in degrees, not the half-angle. |
| θ/2 | The angle whose trigonometric values are calculated. |
| Sine, cosine and tangent | Dimensionless signed ratios determined by the quadrant of θ/2. Tangent is undefined when cosine is zero. |
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.
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.
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.
If you enter 300°, the half-angle is 150°, so cosine and tangent are negative. If the half-angle is 90° plus a multiple of 180°, the page marks tangent undefined instead of displaying a misleading finite number.
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.
Check that both calculators use degrees and the same original angle. This page evaluates the signed function at θ/2, so its sign follows the half-angle’s quadrant.
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.
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.
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.