Enter any angle (θ) in degrees or radians and this Half Angle Calculator computes all three half-angle trig functions — sin(θ/2), cos(θ/2), and tan(θ/2) — using the standard half-angle identities. Choose your angle unit, type in your value, and get the decimal results for all three functions at once.
sin(θ/2)
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cos(θ/2)
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tan(θ/2)
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Half Angle (θ/2)
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A half angle refers to half of a given angle θ, expressed as θ/2. Half-angle identities are trigonometric formulas that express sin(θ/2), cos(θ/2), and tan(θ/2) in terms of the trigonometric functions of the full angle θ. They are derived from the double-angle and power-reducing identities.
The sine of a half angle is found using the formula sin(θ/2) = ±√((1 − cos θ) / 2). The sign (+ or −) depends on which quadrant the half angle θ/2 falls in — positive in the first and second quadrants, negative in the third and fourth.
The cosine half-angle formula is cos(θ/2) = ±√((1 + cos θ) / 2). The tangent half-angle formula is tan(θ/2) = sin θ / (1 + cos θ), which is equivalent to (1 − cos θ) / sin θ. These formulas are standard identities derived from the power-reducing identities.
The sign of the result depends on the quadrant in which the half angle (θ/2) lies. If θ/2 is in the first or second quadrant, sine is positive. Cosine is positive in the first and fourth quadrants. When using the tangent formula sin θ / (1 + cos θ), the sign is automatically determined without needing to check the quadrant.
cos(15°) can be found by treating it as cos(30°/2). Using the half-angle formula: cos(15°) = √((1 + cos 30°) / 2) = √((1 + √3/2) / 2) ≈ 0.9659. Since 15° is in the first quadrant, the result is positive.
No, they are not the same. cos(x/2) refers to the cosine of the half angle, computed using the half-angle identity √((1 + cos x) / 2). cos(x)/2 simply divides the cosine of the full angle by 2. These produce different values for nearly all angles.
Yes. This calculator supports both degrees and radians. Select the unit from the dropdown before entering your angle value. For example, entering π/3 radians (approximately 1.0472) is equivalent to entering 60 degrees.
Half-angle identities are used in calculus for integration of trigonometric expressions, in geometry for computing exact values of angles like 15°, 22.5°, and 67.5°, and in signal processing. They also appear frequently in solving trigonometric equations and simplifying complex expressions.