Calculate sin, cos, and tan of any half angle using half-angle identities. Enter the full angle in degrees for instant exact trigonometric values of the half angle.
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Verified: NIST Digital Library of Mathematical Functions — Trigonometric Identities — April 2026
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Enter the full angle — result will be sin/cos/tan of half this angle
Calculate sin, cos, and tan of any half angle using half-angle identities. Enter the full angle in degrees for instant exact trigonometric values of the half angle.
Half Angle Values
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Sources & Methodology
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Half-angle formulas are trigonometric identities that express sin, cos, and tan of half an angle in terms of the trigonometric functions of the full angle. They are essential in calculus, integration, solving trigonometric equations, and simplifying complex expressions. These identities derive from the double-angle formulas and are fundamental to advanced trigonometry.
The Three Half-Angle Formulas
The half-angle identities are: sin(theta/2) = +/- sqrt((1 - cos theta) / 2), cos(theta/2) = +/- sqrt((1 + cos theta) / 2), tan(theta/2) = sin theta / (1 + cos theta) = (1 - cos theta) / sin theta. The +/- sign depends on the quadrant where theta/2 falls. If theta/2 is in quadrant I or II, sin is positive. If in quadrant I or IV, cos is positive.
Deriving Half-Angle Formulas from Double-Angle Identities
Half-angle formulas come directly from double-angle identities. Starting with cos(2x) = 1 - 2sin^2(x), substitute x = theta/2 and solve: cos(theta) = 1 - 2sin^2(theta/2), so sin^2(theta/2) = (1 - cos theta)/2, and sin(theta/2) = sqrt((1 - cos theta)/2). Similarly from cos(2x) = 2cos^2(x) - 1 we get the cosine half-angle formula.
Common Half-Angle Values to Know
Several common angles produce exact half-angle values. sin(15 degrees) = sin(30/2) = sqrt((1 - sqrt(3)/2)/2) = (sqrt(6) - sqrt(2))/4. cos(22.5 degrees) = cos(45/2) = sqrt((1 + sqrt(2)/2)/2). tan(22.5 degrees) = sqrt(2) - 1. These exact values appear frequently in geometry and calculus problems.
Applications of Half-Angle Formulas
Half-angle formulas appear in integration of trigonometric functions, particularly sin^2(x) = (1 - cos 2x)/2 and cos^2(x) = (1 + cos 2x)/2. These are used to integrate powers of sine and cosine. They also appear in Weierstrass substitution (t = tan(x/2)) which converts rational trig integrands into rational functions.
sin(theta/2) = sqrt((1 - cos theta) / 2) | cos(theta/2) = sqrt((1 + cos theta) / 2) | tan(theta/2) = sin theta / (1 + cos theta)
The sign (+/-) is determined by the quadrant of theta/2. For most standard angle problems, theta/2 falls in Q1 where all trig functions are positive. Worked example: theta = 60 degrees. cos(60) = 0.5. sin(30) = sqrt((1-0.5)/2) = sqrt(0.25) = 0.5. Verified: sin(30 degrees) = 0.5. Correct.
Common Half-Angle Values
Full Angle (theta)
Half Angle (theta/2)
sin(theta/2)
cos(theta/2)
tan(theta/2)
0°
0°
0
1
0
30°
15°
0.2588
0.9659
0.2679
60°
30°
0.5000
0.8660
0.5774
90°
45°
0.7071
0.7071
1.0000
120°
60°
0.8660
0.5000
1.7321
180°
90°
1.0000
0
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💡 Memory Trick: For the half-angle formulas, remember '1 minus for sine, 1 plus for cosine.' sin(theta/2) uses (1 - cos theta), while cos(theta/2) uses (1 + cos theta). Both are divided by 2 under the square root. The tangent formula avoids the square root: tan(theta/2) = sin(theta) / (1 + cos(theta)).
Frequently Asked Questions
There are three half-angle formulas: sin(theta/2) = sqrt((1 - cos theta)/2), cos(theta/2) = sqrt((1 + cos theta)/2), tan(theta/2) = sin(theta)/(1 + cos theta). The sign depends on which quadrant theta/2 falls in.
They derive from double-angle identities. cos(2x) = 1 - 2sin^2(x). Substitute x = theta/2: cos(theta) = 1 - 2sin^2(theta/2). Rearrange: sin^2(theta/2) = (1 - cos theta)/2. Take square root: sin(theta/2) = sqrt((1 - cos theta)/2).
tan(theta/2) = sin(theta)/(1 + cos theta) = (1 - cos theta)/sin(theta). Both forms are equivalent. The advantage of the tangent formula is that it avoids the ambiguous +/- sign of the sine and cosine formulas.
The sign (+/-) depends on the quadrant of theta/2, not theta. If theta/2 is in quadrant I (0-90 degrees), both sin and cos are positive. In Q2 (90-180), sin is positive, cos is negative. In Q3 (180-270), both are negative. In Q4 (270-360), sin is negative, cos is positive.
Half-angle formulas are used to integrate powers of sine and cosine. The formulas sin^2(x) = (1 - cos 2x)/2 and cos^2(x) = (1 + cos 2x)/2 (derived from half-angle identities) convert squared trig functions into forms that can be integrated directly.
The Weierstrass substitution t = tan(theta/2) transforms any rational trigonometric integral into a rational function of t. With this substitution: sin(theta) = 2t/(1+t^2), cos(theta) = (1-t^2)/(1+t^2), d(theta) = 2 dt/(1+t^2). This is a powerful technique for difficult trig integrals.
Half-angle formulas are the inverse of double-angle formulas. The double-angle formula cos(2x) = 1 - 2sin^2(x) becomes the half-angle formula sin(x) = sqrt((1 - cos(2x))/2) when you substitute x = theta/2 and solve for sin(theta/2). They are two sides of the same identity.